%------------------------------------------------------------------------------
% File     : ITP256_2 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Delete 00582_038585
% 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    : 0072_VEBT_Delete_00582_038585 [Des22]

% Status   : Theorem
% Rating   : 0.50 v8.1.0
% Syntax   : Number of formulae    : 11696 (2530 unt;1722 typ;   0 def)
%            Number of atoms       : 30748 (8788 equ)
%            Maximal formula atoms :   73 (   3 avg)
%            Number of connectives : 23735 (2961   ~; 415   |;2769   &)
%                                         (2268 <=>;15322  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   25 (   2 avg)
%            Number of types       :   13 (  12 usr)
%            Number of type conns  : 1461 (1214   >; 247   *;   0   +;   0  <<)
%            Number of predicates  :  264 ( 261 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1449 (1449 usr;  72 con; 0-7 aty)
%            Number of variables   : 37029 (33270   !;1090   ?;37029   :)
%                                         (2669  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 05:02:39.367
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

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

tff(ty_t_Code__Evaluation_Oterm,type,
    code_term: $tType ).

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

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

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

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

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

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

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

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

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

tff(ty_t_HOL_Obool,type,
    bool: $tType ).

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

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

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

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

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

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

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

% Explicit typings (1702)
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_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_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_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_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_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] : ( 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] : fun(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] : fun(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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__alb____,type,
    aTP_Lamp_alb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alc____,type,
    aTP_Lamp_alc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ald____,type,
    aTP_Lamp_ald: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ale____,type,
    aTP_Lamp_ale: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alf____,type,
    aTP_Lamp_alf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alg____,type,
    aTP_Lamp_alg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alh____,type,
    aTP_Lamp_alh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ali____,type,
    aTP_Lamp_ali: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alj____,type,
    aTP_Lamp_alj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alk____,type,
    aTP_Lamp_alk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__all____,type,
    aTP_Lamp_all: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alm____,type,
    aTP_Lamp_alm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aln____,type,
    aTP_Lamp_aln: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alo____,type,
    aTP_Lamp_alo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alp____,type,
    aTP_Lamp_alp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alq____,type,
    aTP_Lamp_alq: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__gk____,type,
    aTP_Lamp_gk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gl____,type,
    aTP_Lamp_gl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_BNF__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__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_Basic__BNF__LFPs_Oprod_Osize__prod,type,
    basic_BNF_size_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * product_prod(A,B) ) > nat ) ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: ( nat * nat ) > nat ).

tff(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int * int ) > int ).

tff(sy_c_Bit__Operations_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 * 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 * 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,bool) ) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > bool ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: ( num * num ) > option(num) ).

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

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
    bit_un2901131394128224187um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
    boolea2506097494486148201lgebra: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A ) > $o ) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
    boolea3799213064322606851m_diff: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A * fun(A,fun(A,A)) ) > $o ) ).

tff(sy_c_COMBB,type,
    combb: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).

tff(sy_c_COMBC,type,
    combc: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).

tff(sy_c_COMBS,type,
    combs: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,bool) ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

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

tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

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

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,bool)) * fun(A,bool) ) > $o ) ).

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

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

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

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

tff(sy_c_Complex_OArg,type,
    arg: complex > real ).

tff(sy_c_Complex_Ocis,type,
    cis: real > complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: ( real * real ) > complex ).

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

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

tff(sy_c_Countable__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_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( product_prod(A,A) > $o ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).

tff(sy_c_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)) * 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_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,bool) * filter(A) ) > $o ) ).

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

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

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

tff(sy_c_Filter_Ofinite__subsets__at__top,type,
    finite5375528669736107172at_top: 
      !>[A: $tType] : ( set(A) > filter(set(A)) ) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * filter(A) ) > filter(B) ) ).

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

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

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),bool) ).

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

tff(sy_c_Finite__Set_Ofold__graph,type,
    finite_fold_graph: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > fun(B,bool) ) ).

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__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_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Omap__fun,type,
    map_fun: 
      !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(fun(A,B),fun(C,D)) ) ).

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

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

tff(sy_c_Fun__Def_Omax__strict,type,
    fun_max_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omax__weak,type,
    fun_max_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omin__strict,type,
    fun_min_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omin__weak,type,
    fun_min_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

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),bool)) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
    groups1962203154675924110t_prod: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(A,fun(list(B),A))) ).

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

tff(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
    groups5270119922927024881d_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_HOL_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

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

tff(sy_c_If,type,
    if: 
      !>[A: $tType] : ( ( bool * 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: fun(product_prod(nat,nat),int) ).

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

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

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

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

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

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

tff(sy_c_Int_Opower__int,type,
    power_int: 
      !>[A: $tType] : ( ( A * int ) > A ) ).

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

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

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
    lattic501386751177426532rg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > fun(B,bool) ) ).

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__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_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_List_OBleast,type,
    bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).

tff(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).

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

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

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

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

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

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

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

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : fun(fun(A,bool),fun(list(A),list(A))) ).

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

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

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

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

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

tff(sy_c_List_Oinsert,type,
    insert: 
      !>[A: $tType] : fun(A,fun(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_Olexn,type,
    lexn: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(nat,set(product_prod(list(A),list(A)))) ) ).

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

tff(sy_c_List_Olexordp,type,
    lexordp: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olinorder_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,bool)) > fun(fun(B,A),fun(B,fun(list(B),list(B)))) ) ).

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

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

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

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

tff(sy_c_List_Olinorder__class_Ostable__sort__key,type,
    linord3483353639454293061rt_key: 
      !>[B: $tType,A: $tType] : ( fun(fun(B,A),fun(list(B),list(B))) > $o ) ).

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

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

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

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

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

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

tff(sy_c_List_Olist_Orec__list,type,
    rec_list: 
      !>[C: $tType,A: $tType] : fun(C,fun(fun(A,fun(list(A),fun(C,C))),fun(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) > fun(list(A),nat) ) ).

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

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

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

tff(sy_c_List_Olistrel1,type,
    listrel1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olistrel1p,type,
    listrel1p: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olistrelp,type,
    listrelp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

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] : fun(list(A),bool) ).

tff(sy_c_List_Oord_Olexordp,type,
    lexordp2: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(list(A),fun(list(A),bool)) ) ).

tff(sy_c_List_Oord__class_Olexordp,type,
    ord_lexordp: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

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

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

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

tff(sy_c_List_Oremdups,type,
    remdups: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremdups__adj,type,
    remdups_adj: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremdups__adj__rel,type,
    remdups_adj_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

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

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

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

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

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

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

tff(sy_c_List_Oset__Cons,type,
    set_Cons: 
      !>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).

tff(sy_c_List_Oshuffles,type,
    shuffles: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).

tff(sy_c_List_Oshuffles__rel,type,
    shuffles_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).

tff(sy_c_List_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > $o ) ).

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

tff(sy_c_List_Osplice__rel,type,
    splice_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_Map_Omap__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_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),bool)) ).

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__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_Onum_OBit0,type,
    bit0: fun(num,num) ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

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

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Order__Relation_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_Orelation__of,type,
    order_relation_of: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > set(product_prod(A,A)) ) ).

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

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

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

tff(sy_c_Orderings_Oord_OLeast,type,
    least: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,bool),A) ) ).

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

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

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

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

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

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oorder__class_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),bool) ).

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

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

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

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

tff(sy_c_Partial__Function_Ofun__ord,type,
    partial_fun_ord: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,bool)),fun(fun(C,A),fun(fun(C,B),bool))) ).

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

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

tff(sy_c_Product__Type_OPair,type,
    product_Pair: 
      !>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).

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

tff(sy_c_Product__Type_Oapfst,type,
    product_apfst: 
      !>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).

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

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

tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).

tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

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

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

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

tff(sy_c_Rat_OFract,type,
    fract: fun(int,fun(int,rat)) ).

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

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

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

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

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

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,bool)) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,bool) ).

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

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_Real_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,bool)) ).

tff(sy_c_Real_Opositive,type,
    positive2: fun(real,bool) ).

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

tff(sy_c_Real_Orep__real,type,
    rep_real: ( real * nat ) > rat ).

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

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Oconstruct,type,
    real_V4425403222259421789struct: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * A ) > B ) ).

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

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

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
    real_V557655796197034286t_dist: 
      !>[A: $tType] : ( ( A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_Oextend__basis,type,
    real_V4986007116245087402_basis: 
      !>[A: $tType] : ( set(A) > set(A) ) ).

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

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
    real_V7770717601297561774m_norm: 
      !>[A: $tType] : ( A > real ) ).

tff(sy_c_Real__Vector__Spaces_Oof__real,type,
    real_Vector_of_real: 
      !>[A: $tType] : ( real > A ) ).

tff(sy_c_Real__Vector__Spaces_Orepresentation,type,
    real_V7696804695334737415tation: 
      !>[A: $tType] : ( ( set(A) * A ) > 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_Relation_OField,type,
    field2: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).

tff(sy_c_Relation_OId,type,
    id: 
      !>[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)) * set(A) ) > set(B) ) ).

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

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,bool)) > $o ) ).

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

tff(sy_c_Relation_Orelcomp,type,
    relcomp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).

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

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

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

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun(bool,A) ).

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

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

tff(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( ( fun(nat,A) * A ) > $o ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,bool),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : fun(set(A),fun(set(A),bool)) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,bool) * 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,bool)) * 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))),bool)) ).

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

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

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

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

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

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_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_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

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

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

tff(sy_c_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

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

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

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

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun(vEBT_VEBT,fun(nat,bool)) ).

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

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

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

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),bool)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,bool) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option(nat) * option(nat) ) > bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option(nat) * option(nat) ) > bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
    vEBT_V2048590022279873568_shift: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(option(A),fun(option(A),option(A))) ) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).

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

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

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

tff(sy_c_Wellfounded_Omax__extp,type,
    max_extp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),fun(set(A),bool)) ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

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

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set(product_prod(nat,nat)) ).

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

tff(sy_c_Wfrec_Osame__fst,type,
    same_fst: 
      !>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

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

tff(sy_c_Zorn_Ochain__subset,type,
    chain_subset: 
      !>[A: $tType] : ( set(set(A)) > $o ) ).

tff(sy_c_Zorn_Ochains,type,
    chains2: 
      !>[A: $tType] : ( set(set(A)) > set(set(set(A))) ) ).

tff(sy_c_Zorn_Oinit__seg__of,type,
    init_seg_of: 
      !>[A: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))) ).

tff(sy_c_Zorn_Opred__on_Ochain,type,
    pred_chain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) ) > fun(set(A),bool) ) ).

tff(sy_c_Zorn_Opred__on_Omaxchain,type,
    pred_maxchain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) * set(A) ) > $o ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : ( fun(A,bool) > bool ) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_fEx,type,
    fEx: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun(bool,bool) ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: ( bool * bool ) > bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_fimplies,type,
    fimplies: fun(bool,fun(bool,bool)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : fun(A,fun(set(A),bool)) ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_maxi____,type,
    maxi: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list(vEBT_VEBT) ).

tff(sy_v_xa____,type,
    xa: nat ).

tff(sy_v_ya____,type,
    ya: nat ).

% Relevant facts (9164)
tff(fact_0__C4_Ohyps_C_I3_J,axiom,
    m = na ).

% "4.hyps"(3)
tff(fact_1_xmi,axiom,
    xa != mi ).

% xmi
tff(fact_2_True,axiom,
    xa = ma ).

% True
tff(fact_3_valid__eq,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
    <=> vEBT_invar_vebt(T2,D2) ) ).

% valid_eq
tff(fact_4_valid__eq1,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(T2,D2)
     => vEBT_VEBT_valid(T2,D2) ) ).

% valid_eq1
tff(fact_5_valid__eq2,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
     => vEBT_invar_vebt(T2,D2) ) ).

% valid_eq2
tff(fact_6_valid__0__not,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% valid_0_not
tff(fact_7_valid__tree__deg__neq__0,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_8__092_060open_062high_Ax_An_A_060_Alength_AtreeList_092_060close_062,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(xa,na)),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList))) ).

% \<open>high x n < length treeList\<close>
tff(fact_9__C4_Ohyps_C_I1_J,axiom,
    vEBT_invar_vebt(summary,m) ).

% "4.hyps"(1)
tff(fact_10_False,axiom,
    ~ pp(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))) ).

% False
tff(fact_11_deg__deg__n,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),N)
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_12_set__vebt__set__vebt_H__valid,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( vEBT_set_vebt(T2) = vEBT_VEBT_set_vebt(T2) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_13_notemp,axiom,
    ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na))),X_1)) ).

% notemp
tff(fact_14_hprolist,axiom,
    aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),vEBT_VEBT_high(xa,na)) = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)) ).

% hprolist
tff(fact_15_not__min__Null__member,axiom,
    ! [T2: vEBT_VEBT] :
      ( ~ pp(vEBT_VEBT_minNull(T2))
     => ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X_1)) ) ).

% not_min_Null_member
tff(fact_16_deg__not__0,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% deg_not_0
tff(fact_17_bit__split__inv,axiom,
    ! [X: nat,D2: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(X,D2),vEBT_VEBT_low(X,D2),D2) = X ).

% bit_split_inv
tff(fact_18__C4_OIH_C_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_delete(summary,X)),Y))
    <=> ( ( X != Y )
        & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,summary),Y)) ) ) ).

% "4.IH"(2)
tff(fact_19_mimapr,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),mi),ma)) ).

% mimapr
tff(fact_20__092_060open_062length_AtreeList_A_061_Alength_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_J_092_060close_062,axiom,
    aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList) = aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))) ).

% \<open>length treeList = length (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)])\<close>
tff(fact_21__C4_Ohyps_C_I7_J,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),mi),ma)) ).

% "4.hyps"(7)
tff(fact_22__C4_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList)))
       => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ).

% "4.hyps"(6)
tff(fact_23__C4_OIH_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList)))
     => ( vEBT_invar_vebt(X2,na)
        & ! [Xa: nat,Xb: nat] :
            ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_delete(X2,Xa)),Xb))
          <=> ( ( Xa != Xb )
              & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),Xb)) ) ) ) ) ).

% "4.IH"(1)
tff(fact_24_in__children__def,axiom,
    ! [N: nat,TreeList: list(vEBT_VEBT),X: nat] :
      ( vEBT_V5917875025757280293ildren(N,TreeList,X)
    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,N))),vEBT_VEBT_low(X,N))) ) ).

% in_children_def
tff(fact_25_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_26_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => vEBT_invar_vebt(vEBT_vebt_buildup(N),N) ) ).

% buildup_gives_valid
tff(fact_27_nth__list__update__neq,axiom,
    ! [A: $tType,I: nat,J: nat,Xs: list(A),X: A] :
      ( ( I != J )
     => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_28_list__update__id,axiom,
    ! [A: $tType,Xs: list(A),I: nat] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),aa(nat,A,nth(A,Xs),I)) = Xs ).

% list_update_id
tff(fact_29_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I: nat,X: A] : aa(list(A),nat,size_size(list(A)),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_30_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A2)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_31_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% neq0_conv
tff(fact_32_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_nat_zero_code
tff(fact_33_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_34__092_060open_062mi_A_092_060le_062_Ax_A_092_060and_062_Ax_A_092_060le_062_Ama_092_060close_062,axiom,
    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),mi),xa))
    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),xa),ma)) ) ).

% \<open>mi \<le> x \<and> x \<le> ma\<close>
tff(fact_35_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( ( I = J )
         => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),J) = X ) )
        & ( ( I != J )
         => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_36_min__in__set__def,axiom,
    ! [Xs: set(nat),X: nat] :
      ( vEBT_VEBT_min_in_set(Xs,X)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X),Xs))
        & ! [X3: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),Xs))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),X3)) ) ) ) ).

% min_in_set_def
tff(fact_37_max__in__set__def,axiom,
    ! [Xs: set(nat),X: nat] :
      ( vEBT_VEBT_max_in_set(Xs,X)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X),Xs))
        & ! [X3: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),Xs))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),X)) ) ) ) ).

% max_in_set_def
tff(fact_38_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),N: nat] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) ) ) ).

% inthall
tff(fact_39_list__update__overwrite,axiom,
    ! [A: $tType,Xs: list(A),I: nat,X: A,Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),I),Y) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),Y) ).

% list_update_overwrite
tff(fact_40_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),zero_zero(A)))
        <=> ( N = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_41_le0,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% le0
tff(fact_42_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A2)) ).

% bot_nat_0.extremum
tff(fact_43_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I))
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X) = Xs ) ) ).

% list_update_beyond
tff(fact_44_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(fun(A,bool),set(A),collect(A),P)))
    <=> pp(aa(A,bool,P,A2)) ) ).

% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_a(set(A),fun(A,bool),A3)) = A3 ).

% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(A,bool,P,X4))
        <=> pp(aa(A,bool,Q,X4)) )
     => ( aa(fun(A,bool),set(A),collect(A),P) = aa(fun(A,bool),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_47_ext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X4: A] : aa(A,B,F2,X4) = aa(A,B,G,X4)
     => ( F2 = G ) ) ).

% ext
tff(fact_48_set__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(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_49_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y2: nat] :
            ( pp(aa(nat,bool,P,Y2))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y2),B2)) )
       => ? [X4: nat] :
            ( pp(aa(nat,bool,P,X4))
            & ! [Y3: nat] :
                ( pp(aa(nat,bool,P,Y3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),X4)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_50_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
      | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% nat_le_linear
tff(fact_51_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_52_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% eq_imp_le
tff(fact_53_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K)) ) ) ).

% le_trans
tff(fact_54_le__refl,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).

% le_refl
tff(fact_55_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ).

% zero_le
tff(fact_56_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_57_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),zero_zero(nat)))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_58_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),zero_zero(nat)))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_59_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% less_eq_nat.simps(1)
tff(fact_60_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F2,I2)),aa(nat,nat,F2,J2))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,F2,I)),aa(nat,nat,F2,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_61_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ( M != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% le_neq_implies_less
tff(fact_62_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_or_eq_imp_le
tff(fact_63_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_64_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_imp_le_nat
tff(fact_65_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_66_ex__least__nat__le,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
            & ! [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K2))
               => ~ pp(aa(nat,bool,P,I3)) )
            & pp(aa(nat,bool,P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_67_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_pos_if_in_set
tff(fact_68_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) )
    <=> ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_69_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),X: A] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I2))) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => pp(aa(A,bool,P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_70_in__set__conv__nth,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( aa(nat,A,nth(A,Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_71_list__ball__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X4)) )
       => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) ) ) ).

% list_ball_nth
tff(fact_72_nth__mem,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,nth(A,Xs),N)),aa(list(A),set(A),set2(A),Xs))) ) ).

% nth_mem
tff(fact_73_set__update__memI,axiom,
    ! [A: $tType,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)))) ) ).

% set_update_memI
tff(fact_74_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_75_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A2: A] :
          ( ! [X4: A] :
              ( ! [Y3: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,Y3)),aa(A,B,F2,X4)))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,X4)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% measure_induct_rule
tff(fact_76_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A2: A] :
          ( ! [X4: A] :
              ( ! [Y3: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,Y3)),aa(A,B,F2,X4)))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,X4)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% measure_induct
tff(fact_77_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,bool),V: fun(A,nat),X: A] :
      ( ! [X4: A] :
          ( ~ pp(aa(A,bool,P,X4))
         => ? [Y3: A] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y3)),aa(A,nat,V,X4)))
              & ~ pp(aa(A,bool,P,Y3)) ) )
     => pp(aa(A,bool,P,X)) ) ).

% infinite_descent_measure
tff(fact_78_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_79_infinite__descent,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa(nat,bool,P,N2))
         => ? [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
              & ~ pp(aa(nat,bool,P,M2)) ) )
     => pp(aa(nat,bool,P,N)) ) ).

% infinite_descent
tff(fact_80_nat__less__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
             => pp(aa(nat,bool,P,M2)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% nat_less_induct
tff(fact_81_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_irrefl_nat
tff(fact_82_less__not__refl3,axiom,
    ! [S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
     => ( S != T2 ) ) ).

% less_not_refl3
tff(fact_83_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_84_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_not_refl
tff(fact_85_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).

% nat_neq_iff
tff(fact_86_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y) )
         => ( X != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_87_neq__if__length__neq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) != aa(list(A),nat,size_size(list(A)),Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_88_Ex__list__of__length,axiom,
    ! [A: $tType,N: nat] :
    ? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = N ).

% Ex_list_of_length
tff(fact_89_list__update__swap,axiom,
    ! [A: $tType,I: nat,I5: nat,Xs: list(A),X: A,X5: A] :
      ( ( I != I5 )
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),I5),X5) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I5),X5)),I),X) ) ) ).

% list_update_swap
tff(fact_90_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_91_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_92_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N),zero_zero(A))) ) ).

% not_less_zero
tff(fact_93_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ( N != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N)) ) ) ).

% gr_zeroI
tff(fact_94_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P: fun(A,bool),X: A] :
      ( ! [X4: A] :
          ( ( aa(A,nat,V,X4) = zero_zero(nat) )
         => pp(aa(A,bool,P,X4)) )
     => ( ! [X4: A] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X4)))
           => ( ~ pp(aa(A,bool,P,X4))
             => ? [Y3: A] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y3)),aa(A,nat,V,X4)))
                  & ~ pp(aa(A,bool,P,Y3)) ) ) )
       => pp(aa(A,bool,P,X)) ) ) ).

% infinite_descent0_measure
tff(fact_95_infinite__descent0,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( ~ pp(aa(nat,bool,P,N2))
             => ? [M2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
                  & ~ pp(aa(nat,bool,P,M2)) ) ) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% infinite_descent0
tff(fact_96_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_97_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_zeroE
tff(fact_98_not__less0,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% not_less0
tff(fact_99_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_100_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% gr0I
tff(fact_101_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),zero_zero(nat))) ).

% bot_nat_0.extremum_strict
tff(fact_102_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys2: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs2)))
             => pp(aa(list(A),bool,P,Ys2)) )
         => pp(aa(list(A),bool,P,Xs2)) )
     => pp(aa(list(A),bool,P,Xs)) ) ).

% length_induct
tff(fact_103_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_104_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,bool))] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
         => ? [X_13: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,I4),X_13)) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
             => pp(aa(A,bool,aa(nat,fun(A,bool),P,I4),aa(nat,A,nth(A,Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_105_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_106_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_107_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_leaf
tff(fact_108_member__correct,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,vEBT_vebt_member(T2),X))
      <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X),vEBT_set_vebt(T2))) ) ) ).

% member_correct
tff(fact_109_maxidef,axiom,
    aa(nat,option(nat),some(nat),maxi) = vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),vEBT_VEBT_high(xa,na))) ).

% maxidef
tff(fact_110_both__member__options__equiv__member,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
      <=> pp(aa(nat,bool,vEBT_vebt_member(T2),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_111_valid__member__both__member__options,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
       => pp(aa(nat,bool,vEBT_vebt_member(T2),X)) ) ) ).

% valid_member_both_member_options
tff(fact_112_VEBT_Oinject_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: list(vEBT_VEBT),Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_113_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_114_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),A2)) ) ).

% dual_order.refl
tff(fact_115_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X)) ) ).

% order_refl
tff(fact_116_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_min_max
tff(fact_117__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxi_O_ASome_Amaxi_A_061_Avebt__maxt_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Maxi: nat] : aa(nat,option(nat),some(nat),Maxi) != vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),vEBT_VEBT_high(xa,na))) ).

% \<open>\<And>thesis. (\<And>maxi. Some maxi = vebt_maxt (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_118_set__vebt__finite,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(set(nat),bool,finite_finite2(nat),vEBT_VEBT_set_vebt(T2))) ) ).

% set_vebt_finite
tff(fact_119_min__Null__member,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( pp(vEBT_VEBT_minNull(T2))
     => ~ pp(aa(nat,bool,vEBT_vebt_member(T2),X)) ) ).

% min_Null_member
tff(fact_120_maxbmo,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),X) )
     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X)) ) ).

% maxbmo
tff(fact_121_both__member__options__def,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
    <=> ( vEBT_V5719532721284313246member(T2,X)
        | vEBT_VEBT_membermima(T2,X) ) ) ).

% both_member_options_def
tff(fact_122_maxt__member,axiom,
    ! [T2: vEBT_VEBT,N: nat,Maxi2: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Maxi2) )
       => pp(aa(nat,bool,vEBT_vebt_member(T2),Maxi2)) ) ) ).

% maxt_member
tff(fact_123_bot__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( bot(C)
     => ! [X: D] : aa(D,C,bot_bot(fun(D,C)),X) = bot_bot(C) ) ).

% bot_apply
tff(fact_124_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(Tree,N)
     => ( pp(aa(nat,bool,vEBT_vebt_member(Tree),X))
       => ( vEBT_V5719532721284313246member(Tree,X)
          | vEBT_VEBT_membermima(Tree,X) ) ) ) ).

% member_valid_both_member_options
tff(fact_125_maxt__corr__help,axiom,
    ! [T2: vEBT_VEBT,N: nat,Maxi2: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Maxi2) )
       => ( pp(aa(nat,bool,vEBT_vebt_member(T2),X))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Maxi2)) ) ) ) ).

% maxt_corr_help
tff(fact_126_maxt__sound,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T2),X)
       => ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),X) ) ) ) ).

% maxt_sound
tff(fact_127_maxt__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),X) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T2),X) ) ) ).

% maxt_corr
tff(fact_128_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(set(A),bool,finite_finite2(A),aa(list(A),set(A),set2(A),Xs))) ).

% List.finite_set
tff(fact_129_pred__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_pred_in_set(Xs,A2,X_1)
     => ( pp(aa(set(nat),bool,finite_finite2(nat),Xs))
       => ~ ? [X2: nat] :
              ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),Xs))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),A2)) ) ) ) ).

% pred_none_empty
tff(fact_130_succ__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_succ_in_set(Xs,A2,X_1)
     => ( pp(aa(set(nat),bool,finite_finite2(nat),Xs))
       => ~ ? [X2: nat] :
              ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X2),Xs))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),X2)) ) ) ) ).

% succ_none_empty
tff(fact_131_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A3: set(nat)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Z),X))
     => ( vEBT_VEBT_min_in_set(A3,Z)
       => ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
         => ? [X_1: nat] : vEBT_is_pred_in_set(A3,X,X_1) ) ) ) ).

% obtain_set_pred
tff(fact_132_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A3: set(nat),B3: set(nat)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Z))
     => ( vEBT_VEBT_max_in_set(A3,Z)
       => ( pp(aa(set(nat),bool,finite_finite2(nat),B3))
         => ( ( A3 = B3 )
           => ? [X_1: nat] : vEBT_is_succ_in_set(A3,X,X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_133_lesseq__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y))
    <=> vEBT_VEBT_lesseq(aa(nat,option(nat),some(nat),X),aa(nat,option(nat),some(nat),Y)) ) ).

% lesseq_shift
tff(fact_134_pred__member,axiom,
    ! [T2: vEBT_VEBT,X: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T2),X,Y)
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(T2),Y))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X))
        & ! [Z2: nat] :
            ( ( pp(aa(nat,bool,vEBT_vebt_member(T2),Z2))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Z2),X)) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Z2),Y)) ) ) ) ).

% pred_member
tff(fact_135_succ__member,axiom,
    ! [T2: vEBT_VEBT,X: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T2),X,Y)
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(T2),Y))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
        & ! [Z2: nat] :
            ( ( pp(aa(nat,bool,vEBT_vebt_member(T2),Z2))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Z2)) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),Z2)) ) ) ) ).

% succ_member
tff(fact_136_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_137_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),bot_bot(A)),A2)) ) ).

% bot.extremum
tff(fact_138_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),bot_bot(A)))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_139_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),bot_bot(A)))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_140_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),bot_bot(A))) ) ).

% bot.extremum_strict
tff(fact_141_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( ( A2 != bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),A2)) ) ) ).

% bot.not_eq_extremum
tff(fact_142_finite__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A3 ) ).

% finite_list
tff(fact_143_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B3))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3)) ) ) ).

% subset_code(1)
tff(fact_144_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_145_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),X: A,I: nat] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X))),A3)) ) ) ).

% set_update_subsetI
tff(fact_146_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_147_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y)) )
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
               => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z))
                   => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X)) )
                 => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X))
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_148_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% order_class.order_eq_iff
tff(fact_149_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% ord_eq_le_trans
tff(fact_150_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% ord_le_eq_trans
tff(fact_151_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_152_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% order.trans
tff(fact_153_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) ) ) ) ).

% order_trans
tff(fact_154_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A2: A,B2: A] :
          ( ! [A4: A,B4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B4)) )
         => ( ! [A4: A,B4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),P,B4),A4))
               => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B4)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),P,A2),B2)) ) ) ) ).

% linorder_wlog
tff(fact_155_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% dual_order.eq_iff
tff(fact_156_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_157_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% dual_order.trans
tff(fact_158_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_159_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X))) ) ) ).

% le_funD
tff(fact_160_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X))) ) ) ).

% le_funE
tff(fact_161_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X4: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4)))
         => pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G)) ) ) ).

% le_funI
tff(fact_162_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G))
        <=> ! [X3: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3))) ) ) ).

% le_fun_def
tff(fact_163_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_164_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X4: B,Y2: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X4),Y2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_subst1
tff(fact_165_order__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,B2)),C2))
           => ( ! [X4: A,Y2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,X4)),aa(A,C,F2,Y2))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_subst2
tff(fact_166_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_eq_refl
tff(fact_167_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_linear
tff(fact_168_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X4: B,Y2: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X4),Y2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_169_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X4: A,Y2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y2))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,A2)),C2)) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_170_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_le_cases
tff(fact_171_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_172_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X)) ) ).

% lt_ex
tff(fact_173_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X_1)) ) ).

% gt_ex
tff(fact_174_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ? [Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),Y)) ) ) ) ).

% dense
tff(fact_175_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_176_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% order.asym
tff(fact_177_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% ord_eq_less_trans
tff(fact_178_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% ord_less_eq_trans
tff(fact_179_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A2: A] :
          ( ! [X4: A] :
              ( ! [Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X4))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,X4)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% less_induct
tff(fact_180_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_181_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ( X != Y )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_cases
tff(fact_182_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% dual_order.asym
tff(fact_183_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),A2)) ) ).

% dual_order.irrefl
tff(fact_184_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
        <=> ? [N3: A] :
              ( pp(aa(A,bool,P,N3))
              & ! [M3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M3),N3))
                 => ~ pp(aa(A,bool,P,M3)) ) ) ) ) ).

% exists_least_iff
tff(fact_185_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A2: A,B2: A] :
          ( ! [A4: A,B4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A4),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B4)) )
         => ( ! [A4: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,A4),A4))
           => ( ! [A4: A,B4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),P,B4),A4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),P,A4),B4)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A2),B2)) ) ) ) ) ).

% linorder_less_wlog
tff(fact_186_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% order.strict_trans
tff(fact_187_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_188_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% dual_order.strict_trans
tff(fact_189_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_190_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_191_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE
tff(fact_192_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_asym
tff(fact_193_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neq_iff
tff(fact_194_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% order_less_asym'
tff(fact_195_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_less_trans
tff(fact_196_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X4: B,Y2: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X4),Y2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X4)),aa(B,A,F2,Y2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_197_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X4: A,Y2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y2))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,A2)),C2)) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_198_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).

% order_less_irrefl
tff(fact_199_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X4: B,Y2: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X4),Y2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X4)),aa(B,A,F2,Y2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_less_subst1
tff(fact_200_order__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,B2)),C2))
           => ( ! [X4: A,Y2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,X4)),aa(A,C,F2,Y2))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_less_subst2
tff(fact_201_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_not_sym
tff(fact_202_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
           => pp(P) ) ) ) ).

% order_less_imp_triv
tff(fact_203_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          | ( X = Y )
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_less_linear
tff(fact_204_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_205_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_206_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_imp_not_less
tff(fact_207_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% leD
tff(fact_208_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% leI
tff(fact_209_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_210_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_211_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_212_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X4)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ).

% dense_ge
tff(fact_213_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Z)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ).

% dense_le
tff(fact_214_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% less_le_not_le
tff(fact_215_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% not_le_imp_less
tff(fact_216_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_217_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_218_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% order.strict_trans1
tff(fact_219_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% order.strict_trans2
tff(fact_220_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% order.strict_iff_not
tff(fact_221_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X))
         => ( ! [W: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),W))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),X))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),W)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ) ).

% dense_ge_bounded
tff(fact_222_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ! [W: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),W))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ) ).

% dense_le_bounded
tff(fact_223_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_224_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_225_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% dual_order.strict_trans1
tff(fact_226_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% dual_order.strict_trans2
tff(fact_227_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_228_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% order.strict_implies_order
tff(fact_229_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% dual_order.strict_implies_order
tff(fact_230_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_231_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_232_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_not_le
tff(fact_233_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_not_less
tff(fact_234_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_less_imp_le
tff(fact_235_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( ( A2 != B2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% order_le_neq_trans
tff(fact_236_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% order_neq_le_trans
tff(fact_237_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_le_less_trans
tff(fact_238_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_less_le_trans
tff(fact_239_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X4: B,Y2: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X4),Y2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X4)),aa(B,A,F2,Y2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_le_less_subst1
tff(fact_240_order__le__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,B2)),C2))
           => ( ! [X4: A,Y2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,X4)),aa(A,C,F2,Y2))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_le_less_subst2
tff(fact_241_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X4: B,Y2: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X4),Y2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,F2,Y2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,C2))) ) ) ) ) ).

% order_less_le_subst1
tff(fact_242_order__less__le__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F2,B2)),C2))
           => ( ! [X4: A,Y2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,X4)),aa(A,C,F2,Y2))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F2,A2)),C2)) ) ) ) ) ).

% order_less_le_subst2
tff(fact_243_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_le_less_linear
tff(fact_244_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_245_greater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X))
    <=> pp(vEBT_VEBT_greater(aa(nat,option(nat),some(nat),X),aa(nat,option(nat),some(nat),Y))) ) ).

% greater_shift
tff(fact_246_less__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
    <=> pp(vEBT_VEBT_less(aa(nat,option(nat),some(nat),X),aa(nat,option(nat),some(nat),Y))) ) ).

% less_shift
tff(fact_247_mint__sound,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T2),X)
       => ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),X) ) ) ) ).

% mint_sound
tff(fact_248_mint__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),X) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T2),X) ) ) ).

% mint_corr
tff(fact_249_maxt__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_maxt(T2) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T2) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_250_mint__corr__help,axiom,
    ! [T2: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Mini) )
       => ( pp(aa(nat,bool,vEBT_vebt_member(T2),X))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mini),X)) ) ) ) ).

% mint_corr_help
tff(fact_251_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X6: set(A)] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => ? [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),X6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Xa)) ) )
           => ~ pp(aa(set(A),bool,finite_finite2(A),X6)) ) ) ) ).

% infinite_growing
tff(fact_252_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                & ~ ? [Xa: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),S2))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa),X4)) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_253_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X4))
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_254_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa))
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_255_mint__member,axiom,
    ! [T2: vEBT_VEBT,N: nat,Maxi2: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Maxi2) )
       => pp(aa(nat,bool,vEBT_vebt_member(T2),Maxi2)) ) ) ).

% mint_member
tff(fact_256_option_Oinject,axiom,
    ! [A: $tType,X22: A,Y22: A] :
      ( ( aa(A,option(A),some(A),X22) = aa(A,option(A),some(A),Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_257_minminNull,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T2) = none(nat) )
     => pp(vEBT_VEBT_minNull(T2)) ) ).

% minminNull
tff(fact_258_minNullmin,axiom,
    ! [T2: vEBT_VEBT] :
      ( pp(vEBT_VEBT_minNull(T2))
     => ( vEBT_vebt_mint(T2) = none(nat) ) ) ).

% minNullmin
tff(fact_259_finite__code,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A3: set(A)] : pp(aa(set(A),bool,finite_finite2(A),A3)) ) ).

% finite_code
tff(fact_260_mint__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_mint(T2) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T2) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_261_not__Some__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ! [Y4: A] : X != aa(A,option(A),some(A),Y4)
    <=> ( X = none(A) ) ) ).

% not_Some_eq
tff(fact_262_not__None__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ( X != none(A) )
    <=> ? [Y4: A] : X = aa(A,option(A),some(A),Y4) ) ).

% not_None_eq
tff(fact_263_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F2),G))
        <=> ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F2),G))
            & ~ pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),G),F2)) ) ) ) ).

% less_fun_def
tff(fact_264_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,X: option(A),P: fun(option(A),fun(option(B),bool)),Y: option(B)] :
      ( ( ( X = none(A) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) )
     => ( ( ( Y = none(B) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) )
       => ( ! [A4: A,B4: B] :
              ( ( X = aa(A,option(A),some(A),A4) )
             => ( ( Y = aa(B,option(B),some(B),B4) )
               => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) ) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_265_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ! [X_13: option(A)] : pp(aa(option(A),bool,P,X_13))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        & ! [X3: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X3))) ) ) ).

% split_option_all
tff(fact_266_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ? [X_13: option(A)] : pp(aa(option(A),bool,P,X_13))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        | ? [X3: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X3))) ) ) ).

% split_option_ex
tff(fact_267_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X23: A] : Y != aa(A,option(A),some(A),X23) ) ).

% option.exhaust
tff(fact_268_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X22: A] :
      ( ( Option = aa(A,option(A),some(A),X22) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_269_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X22: A] : none(A) != aa(A,option(A),some(A),X22) ).

% option.distinct(1)
tff(fact_270_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_271_finite__maxlen,axiom,
    ! [A: $tType,M4: set(list(A))] :
      ( pp(aa(set(list(A)),bool,finite_finite2(list(A)),M4))
     => ? [N2: nat] :
        ! [X2: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X2),M4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X2)),N2)) ) ) ).

% finite_maxlen
tff(fact_272_option_Osize__neq,axiom,
    ! [A: $tType,X: option(A)] : aa(option(A),nat,size_size(option(A)),X) != zero_zero(nat) ).

% option.size_neq
tff(fact_273_finite__set__choice,axiom,
    ! [B: $tType,A: $tType,A3: set(A),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => ? [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,X4),X_12)) )
       => ? [F3: fun(A,B)] :
          ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,X2),aa(A,B,F3,X2))) ) ) ) ).

% finite_set_choice
tff(fact_274_finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A3: set(A)] : pp(aa(set(A),bool,finite_finite2(A),A3)) ) ).

% finite
tff(fact_275_finite__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,finite_finite2(A),B3))
       => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% finite_subset
tff(fact_276_infinite__super,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),T3))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
       => ~ pp(aa(set(A),bool,finite_finite2(A),T3)) ) ) ).

% infinite_super
tff(fact_277_rev__finite__subset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% rev_finite_subset
tff(fact_278_finite__psubset__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ! [A5: set(A)] :
            ( pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( ! [B5: set(A)] :
                  ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B5),A5))
                 => pp(aa(set(A),bool,P,B5)) )
             => pp(aa(set(A),bool,P,A5)) ) )
       => pp(aa(set(A),bool,P,A3)) ) ) ).

% finite_psubset_induct
tff(fact_279_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => ? [X4: A] :
          ( pp(aa(A,bool,P,X4))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,X4)),aa(A,nat,M,Y3))) ) ) ) ).

% ex_has_least_nat
tff(fact_280_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),A2))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X4))
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_281_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa))
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_282_infinite__imp__nonempty,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
     => ( S2 != bot_bot(set(A)) ) ) ).

% infinite_imp_nonempty
tff(fact_283_finite_OemptyI,axiom,
    ! [A: $tType] : pp(aa(set(A),bool,finite_finite2(A),bot_bot(set(A)))) ).

% finite.emptyI
tff(fact_284_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F2: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [Y2: A] :
            ( pp(aa(A,bool,P,Y2))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,Y2)),B2)) )
       => ? [X4: A] :
            ( pp(aa(A,bool,P,X4))
            & ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,Y3)),aa(A,nat,F2,X4))) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_285_pred__correct,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_pred(T2,X) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T2),X,Sx) ) ) ).

% pred_correct
tff(fact_286_succ__correct,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_succ(T2,X) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(T2),X,Sx) ) ) ).

% succ_correct
tff(fact_287_pred__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_pred(T2,X) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T2),X,Px) ) ) ).

% pred_corr
tff(fact_288_succ__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_succ(T2,X) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T2),X,Sx) ) ) ).

% succ_corr
tff(fact_289_subset__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),bot_bot(set(A))))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_290_empty__subsetI,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),bot_bot(set(A))),A3)) ).

% empty_subsetI
tff(fact_291_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),X: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,X,Y)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Y),Xs))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
        & ! [X3: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),Xs))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),X3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X3)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_292_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),X: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,X,Y)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Y),Xs))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X))
        & ! [X3: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),Xs))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),X))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_293_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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi = Ma )
       => ( ! [X2: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
             => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) )
          & ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_12)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_294_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_295_psubsetI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( ( A3 != B3 )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3)) ) ) ).

% psubsetI
tff(fact_296_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),bot_bot(set(A)))) ).

% empty_iff
tff(fact_297_all__not__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X3: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_298_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( aa(fun(A,bool),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ! [X3: A] : ~ pp(aa(A,bool,P,X3)) ) ).

% Collect_empty_eq
tff(fact_299_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),P) )
    <=> ! [X3: A] : ~ pp(aa(A,bool,P,X3)) ) ).

% empty_Collect_eq
tff(fact_300_subsetI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% subsetI
tff(fact_301_subset__antisym,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => ( A3 = B3 ) ) ) ).

% subset_antisym
tff(fact_302_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Ma),X))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_303_psubsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% psubsetD
tff(fact_304_psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B3),C3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),C3)) ) ) ).

% psubset_trans
tff(fact_305_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Va) = none(nat) ).

% vebt_succ.simps(3)
tff(fact_306_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list(vEBT_VEBT),Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Va),Vb) = none(nat) ).

% vebt_pred.simps(4)
tff(fact_307_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),X)) ).

% vebt_member.simps(2)
tff(fact_308_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : pp(vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy))) ).

% VEBT_internal.minNull.simps(4)
tff(fact_309_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Uu) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) ).

% vebt_delete.simps(4)
tff(fact_310_vebt__pred_Osimps_I5_J,axiom,
    ! [V2: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd,Ve),Vf) = none(nat) ).

% vebt_pred.simps(5)
tff(fact_311_vebt__succ_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc,Vd),Ve) = none(nat) ).

% vebt_succ.simps(4)
tff(fact_312_not__psubset__empty,axiom,
    ! [A: $tType,A3: set(A)] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),bot_bot(set(A)))) ).

% not_psubset_empty
tff(fact_313_emptyE,axiom,
    ! [A: $tType,A2: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),bot_bot(set(A)))) ).

% emptyE
tff(fact_314_equals0D,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ( A3 = bot_bot(set(A)) )
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ).

% equals0D
tff(fact_315_equals0I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [Y2: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
     => ( A3 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_316_ex__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ? [X3: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_317_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) ).

% bot_set_def
tff(fact_318_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_319_in__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3)) ) ) ).

% in_mono
tff(fact_320_subsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% subsetD
tff(fact_321_equalityE,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ) ).

% equalityE
tff(fact_322_subset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3)) ) ) ).

% subset_eq
tff(fact_323_equalityD1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% equalityD1
tff(fact_324_equalityD2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ).

% equalityD2
tff(fact_325_subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> ! [T4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),T4),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),T4),B3)) ) ) ).

% subset_iff
tff(fact_326_subset__refl,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),A3)) ).

% subset_refl
tff(fact_327_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(A,bool,P,X4))
         => pp(aa(A,bool,Q,X4)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(fun(A,bool),set(A),collect(A),Q))) ) ).

% Collect_mono
tff(fact_328_subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3)) ) ) ).

% subset_trans
tff(fact_329_set__eq__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ) ).

% set_eq_subset
tff(fact_330_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(fun(A,bool),set(A),collect(A),Q)))
    <=> ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(A,bool,Q,X3)) ) ) ).

% Collect_mono_iff
tff(fact_331_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
        | ( A3 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_332_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B3),C3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),C3)) ) ) ).

% subset_psubset_trans
tff(fact_333_subset__not__subset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
        & ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ) ).

% subset_not_subset_eq
tff(fact_334_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),C3)) ) ) ).

% psubset_subset_trans
tff(fact_335_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% psubset_imp_subset
tff(fact_336_psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
        & ( A3 != B3 ) ) ) ).

% psubset_eq
tff(fact_337_psubsetE,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ) ).

% psubsetE
tff(fact_338_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list(vEBT_VEBT),Vb: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va,Vb),X)
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_339_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT,Uz: nat] : ~ vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_340_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : ~ pp(vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va,Vb,Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_341_tvalid,axiom,
    vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary),deg) ).

% tvalid
tff(fact_342_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Mi) ).

% vebt_mint.simps(3)
tff(fact_343_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Ma) ).

% vebt_maxt.simps(3)
tff(fact_344_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = none(nat) ).

% vebt_mint.simps(2)
tff(fact_345_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = none(nat) ).

% vebt_maxt.simps(2)
tff(fact_346_calculation,axiom,
    ( ( ya = mi )
   => ( ( xa != ya )
      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary)),ya)) ) ) ).

% calculation
tff(fact_347_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_348_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6) )
    <=> ( ( A2 = A6 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
tff(fact_349_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S2))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N3))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),S2)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_350_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),N4))
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),N4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),M3)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_351_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set(nat)] :
      ( ! [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),M5))
         => ? [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N5))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N5),S2)) ) )
     => ~ pp(aa(set(nat),bool,finite_finite2(nat),S2)) ) ).

% unbounded_k_infinite
tff(fact_352__C4_Ohyps_C_I4_J,axiom,
    deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) ).

% "4.hyps"(4)
tff(fact_353_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F3: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc: A] : X != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B4),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_354_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,A)),Uv2: option(A)] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,A)),V3: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uw2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F3: fun(A,fun(A,A)),A4: A,B4: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),F3),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),A4)),aa(A,option(A),some(A),B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_355_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,bool)),Uv2: option(A)] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,bool)),V3: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),Uw2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F3: fun(A,fun(A,bool)),X4: A,Y2: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),F3),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),X4)),aa(A,option(A),some(A),Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_356_prod__induct7,axiom,
    ! [G2: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))))] :
      ( ! [A4: A,B4: B,C4: C,D3: D,E2: E,F3: F,G3: G2] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G2)))),C4),aa(product_prod(E,product_prod(F,G2)),product_prod(D,product_prod(E,product_prod(F,G2))),aa(D,fun(product_prod(E,product_prod(F,G2)),product_prod(D,product_prod(E,product_prod(F,G2)))),product_Pair(D,product_prod(E,product_prod(F,G2))),D3),aa(product_prod(F,G2),product_prod(E,product_prod(F,G2)),aa(E,fun(product_prod(F,G2),product_prod(E,product_prod(F,G2))),product_Pair(E,product_prod(F,G2)),E2),aa(G2,product_prod(F,G2),aa(F,fun(G2,product_prod(F,G2)),product_Pair(F,G2),F3),G3))))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))))),bool,P,X)) ) ).

% prod_induct7
tff(fact_357_prod__induct6,axiom,
    ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ( ! [A4: A,B4: B,C4: C,D3: D,E2: E,F3: F] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B4),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C4),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D3),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F3)))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),bool,P,X)) ) ).

% prod_induct6
tff(fact_358_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ( ! [A4: A,B4: B,C4: C,D3: D,E2: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C4),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D3),E2))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P,X)) ) ).

% prod_induct5
tff(fact_359_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool),X: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ( ! [A4: A,B4: B,C4: C,D3: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C4),D3)))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P,X)) ) ).

% prod_induct4
tff(fact_360_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,C)),bool),X: product_prod(A,product_prod(B,C))] :
      ( ! [A4: A,B4: B,C4: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C4))))
     => pp(aa(product_prod(A,product_prod(B,C)),bool,P,X)) ) ).

% prod_induct3
tff(fact_361_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))))] :
      ~ ! [A4: A,B4: B,C4: C,D3: D,E2: E,F3: F,G3: G2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G2)))),C4),aa(product_prod(E,product_prod(F,G2)),product_prod(D,product_prod(E,product_prod(F,G2))),aa(D,fun(product_prod(E,product_prod(F,G2)),product_prod(D,product_prod(E,product_prod(F,G2)))),product_Pair(D,product_prod(E,product_prod(F,G2))),D3),aa(product_prod(F,G2),product_prod(E,product_prod(F,G2)),aa(E,fun(product_prod(F,G2),product_prod(E,product_prod(F,G2))),product_Pair(E,product_prod(F,G2)),E2),aa(G2,product_prod(F,G2),aa(F,fun(G2,product_prod(F,G2)),product_Pair(F,G2),F3),G3)))))) ).

% prod_cases7
tff(fact_362_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ~ ! [A4: A,B4: B,C4: C,D3: D,E2: E,F3: F] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B4),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C4),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D3),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F3))))) ).

% prod_cases6
tff(fact_363_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ~ ! [A4: A,B4: B,C4: C,D3: D,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C4),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D3),E2)))) ).

% prod_cases5
tff(fact_364_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A4: A,B4: B,C4: C,D3: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C4),D3))) ).

% prod_cases4
tff(fact_365_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A4: A,B4: B,C4: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C4)) ).

% prod_cases3
tff(fact_366_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6) )
     => ~ ( ( A2 = A6 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
tff(fact_367_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool),P2: product_prod(A,B)] :
      ( ! [A4: A,B4: B] : pp(aa(product_prod(A,B),bool,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)))
     => pp(aa(product_prod(A,B),bool,P,P2)) ) ).

% prod_cases
tff(fact_368_surj__pair,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B)] :
    ? [X4: A,Y2: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2) ).

% surj_pair
tff(fact_369_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A4: A,B4: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) ).

% old.prod.exhaust
tff(fact_370_bounded__Max__nat,axiom,
    ! [P: fun(nat,bool),X: nat,M4: nat] :
      ( pp(aa(nat,bool,P,X))
     => ( ! [X4: nat] :
            ( pp(aa(nat,bool,P,X4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),M4)) )
       => ~ ! [M5: nat] :
              ( pp(aa(nat,bool,P,M5))
             => ~ ! [X2: nat] :
                    ( pp(aa(nat,bool,P,X2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),M5)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_371_finite__transitivity__chain,axiom,
    ! [A: $tType,A3: set(A),R: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ! [X4: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),R,X4),X4))
       => ( ! [X4: A,Y2: A,Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),R,X4),Y2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),R,Y2),Z3))
               => pp(aa(A,bool,aa(A,fun(A,bool),R,X4),Z3)) ) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => ? [Y3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),R,X4),Y3)) ) )
           => ( A3 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_372_finite__nat__set__iff__bounded,axiom,
    ! [N4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),N4))
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),N4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),M3)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_373_infinite__nat__iff__unbounded,axiom,
    ! [S2: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S2))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N3))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),S2)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_374_bounded__nat__set__is__finite,axiom,
    ! [N4: set(nat),N: nat] :
      ( ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),N4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),N)) )
     => pp(aa(set(nat),bool,finite_finite2(nat),N4)) ) ).

% bounded_nat_set_is_finite
tff(fact_375_old_Oprod_Orec,axiom,
    ! [A: $tType,T: $tType,B: $tType,F1: fun(A,fun(B,T)),A2: A,B2: B] : product_rec_prod(A,B,T,F1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) = aa(B,T,aa(A,fun(B,T),F1,A2),B2) ).

% old.prod.rec
tff(fact_376_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ( aa(fun(A,bool),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ( P = bot_bot(fun(A,bool)) ) ) ).

% Collect_empty_eq_bot
tff(fact_377_bot__empty__eq,axiom,
    ! [A: $tType,X2: A] :
      ( pp(aa(A,bool,bot_bot(fun(A,bool)),X2))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),bot_bot(set(A)))) ) ).

% bot_empty_eq
tff(fact_378_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ~ ? [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S2))
                  & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S2)))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_379_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),Y: A,F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S2))),aa(A,B,F2,Y))) ) ) ) ) ).

% arg_min_least
tff(fact_380_nat__descend__induct,axiom,
    ! [N: nat,P: fun(nat,bool),M: nat] :
      ( ! [K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
         => pp(aa(nat,bool,P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
           => ( ! [I3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),I3))
                 => pp(aa(nat,bool,P,I3)) )
             => pp(aa(nat,bool,P,K2)) ) )
       => pp(aa(nat,bool,P,M)) ) ) ).

% nat_descend_induct
tff(fact_381_subset__emptyI,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X4: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),bot_bot(set(A)))) ) ).

% subset_emptyI
tff(fact_382_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D1))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D22))
           => ? [E2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),E2),D1))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),E2),D22)) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_383_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),zero_zero(A))) ) ).

% less_numeral_extra(3)
tff(fact_384_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,P,A2))
           => ( ~ pp(aa(A,bool,P,B2))
             => ? [C4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C4),B2))
                  & ! [X2: A] :
                      ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X2))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),C4)) )
                     => pp(aa(A,bool,P,X2)) )
                  & ! [D4: A] :
                      ( ! [X4: A] :
                          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
                            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),D4)) )
                         => pp(aa(A,bool,P,X4)) )
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D4),C4)) ) ) ) ) ) ) ).

% complete_interval
tff(fact_385_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B6: B,A6: B] :
          ( ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B6),A6))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A6),B6)) ) ) ).

% verit_comp_simplify1(3)
tff(fact_386_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_387_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_388_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% add_le_cancel_left
tff(fact_389_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% add_le_cancel_right
tff(fact_390_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.right_neutral
tff(fact_391_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_392_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_393_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_394_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_395_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_396_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_397_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_398_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add_0
tff(fact_399_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% add_less_cancel_left
tff(fact_400_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% add_less_cancel_right
tff(fact_401_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_402_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_403_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_404_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_405_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_406_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_407_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel2
tff(fact_408_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel1
tff(fact_409_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% add_le_same_cancel2
tff(fact_410_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% add_le_same_cancel1
tff(fact_411_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% add_less_same_cancel1
tff(fact_412_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% add_less_same_cancel2
tff(fact_413_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel1
tff(fact_414_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel2
tff(fact_415_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_416_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_417_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% add_gr_0
tff(fact_418_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_419_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_420_add_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.left_commute
tff(fact_421_add_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) ) ).

% add.commute
tff(fact_422_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_423_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_424_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.assoc
tff(fact_425_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B3: A,K: A,B2: A,A2: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_426_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_427_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_428_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_429_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% verit_sum_simplify
tff(fact_430_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
            & ( K = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_431_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_432_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_433_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_mono
tff(fact_434_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% add_left_mono
tff(fact_435_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ~ ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) ) ) ).

% less_eqE
tff(fact_436_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).

% add_right_mono
tff(fact_437_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ? [C5: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C5) ) ) ).

% le_iff_add
tff(fact_438_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% add_le_imp_le_left
tff(fact_439_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% add_le_imp_le_right
tff(fact_440_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% comm_monoid_add_class.add_0
tff(fact_441_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.comm_neutral
tff(fact_442_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add.group_left_neutral
tff(fact_443_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_444_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_445_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
            & ( K = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_446_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_strict_mono
tff(fact_447_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% add_strict_left_mono
tff(fact_448_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).

% add_strict_right_mono
tff(fact_449_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% add_less_imp_less_left
tff(fact_450_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% add_less_imp_less_right
tff(fact_451_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).

% plus_nat.add_0
tff(fact_452_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = M )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_453_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K)) ) ).

% add_lessD1
tff(fact_454_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_less_mono
tff(fact_455_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I)) ).

% not_add_less1
tff(fact_456_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I)) ).

% not_add_less2
tff(fact_457_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_less_mono1
tff(fact_458_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_less_add1
tff(fact_459_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_less_add2
tff(fact_460_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% less_add_eq_less
tff(fact_461_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).

% add_leE
tff(fact_462_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% le_add1
tff(fact_463_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ).

% le_add2
tff(fact_464_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% add_leD1
tff(fact_465_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).

% add_leD2
tff(fact_466_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
     => ? [N2: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2) ) ).

% le_Suc_ex
tff(fact_467_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_le_mono
tff(fact_468_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_le_mono1
tff(fact_469_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_le_add1
tff(fact_470_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_le_add2
tff(fact_471_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_472_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_473_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_474_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_nonpos
tff(fact_475_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_nonneg_nonneg
tff(fact_476_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_increasing2
tff(fact_477_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2)) ) ) ) ).

% add_decreasing2
tff(fact_478_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_increasing
tff(fact_479_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2)) ) ) ) ).

% add_decreasing
tff(fact_480_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_less_le_mono
tff(fact_481_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_le_less_mono
tff(fact_482_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_483_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_484_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_neg_neg
tff(fact_485_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_pos_pos
tff(fact_486_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ ! [C4: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) )
               => ( C4 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_487_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% pos_add_strict
tff(fact_488_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ? [K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_489_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M: nat,K: nat] :
      ( ! [M5: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F2,M5)),aa(nat,nat,F2,N2))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,M)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_490_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F2: fun(A,nat),N: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
           => ? [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
                & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,Y3)),aa(A,nat,F2,X4))) ) )
       => ? [Y2: A] :
            ( pp(aa(A,bool,P,Y2))
            & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,Y2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),N))) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_491_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_neg_nonpos
tff(fact_492_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_nonneg_pos
tff(fact_493_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_neg
tff(fact_494_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_pos_nonneg
tff(fact_495_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_strict_increasing
tff(fact_496_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_strict_increasing2
tff(fact_497_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),A2)) ) ).

% verit_comp_simplify1(2)
tff(fact_498_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% verit_la_disequality
tff(fact_499_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),A2)) ) ).

% verit_comp_simplify1(1)
tff(fact_500_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A2: A] :
        ? [B4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B4))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),A2)) ) ) ).

% ex_gt_or_lt
tff(fact_501_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X4: A,Y2: B] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2)),R2))
         => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2)),S)) )
     => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R2),S)) ) ).

% subrelI
tff(fact_502_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R2: A,S: B,R: set(product_prod(A,B)),S3: B] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S)),R))
     => ( ( S3 = S )
       => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S3)),R)) ) ) ).

% ssubst_Pair_rhs
tff(fact_503_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),lattic7623131987881927897min_on(A,B,F2,S2)),S2)) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_504_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),zero_zero(A))) ) ).

% le_numeral_extra(3)
tff(fact_505_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_506_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(nat,option(nat),some(nat),X)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% add_shift
tff(fact_507_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_epsilon
tff(fact_508_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A))) ) ) ) ).

% add_less_zeroD
tff(fact_509_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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,N,Xs)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_510_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),B4))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,B4),A4)) )
     => ( ! [A4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),zero_zero(nat)))
       => ( ! [A4: nat,B4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),B4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B4))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A2),B2)) ) ) ) ).

% Euclid_induct
tff(fact_511_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A2: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_512_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X2)) ) ) ).

% minf(8)
tff(fact_513_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),T2)) ) ) ).

% minf(6)
tff(fact_514_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X2)) ) ) ).

% pinf(8)
tff(fact_515_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),T2)) ) ) ).

% pinf(6)
tff(fact_516_length__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,N,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_enumerate
tff(fact_517_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_518_less__by__empty,axiom,
    ! [A: $tType,A3: set(product_prod(A,A)),B3: set(product_prod(A,A))] :
      ( ( A3 = bot_bot(set(product_prod(A,A))) )
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),A3),B3)) ) ).

% less_by_empty
tff(fact_519_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_520_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X2: A] :
        ? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),X_1)) ) ).

% linordered_field_no_ub
tff(fact_521_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X2: A] :
        ? [Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X2)) ) ).

% linordered_field_no_lb
tff(fact_522_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [Z4: A] :
            ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X4))
             => ( pp(aa(A,bool,P,X4))
              <=> pp(aa(A,bool,P3,X4)) ) )
         => ( ? [Z4: A] :
              ! [X4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X4))
               => ( pp(aa(A,bool,Q,X4))
                <=> pp(aa(A,bool,Q2,X4)) ) )
           => ? [Z3: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X2))
               => ( ( pp(aa(A,bool,P,X2))
                    & pp(aa(A,bool,Q,X2)) )
                <=> ( pp(aa(A,bool,P3,X2))
                    & pp(aa(A,bool,Q2,X2)) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_523_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [Z4: A] :
            ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X4))
             => ( pp(aa(A,bool,P,X4))
              <=> pp(aa(A,bool,P3,X4)) ) )
         => ( ? [Z4: A] :
              ! [X4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X4))
               => ( pp(aa(A,bool,Q,X4))
                <=> pp(aa(A,bool,Q2,X4)) ) )
           => ? [Z3: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X2))
               => ( ( pp(aa(A,bool,P,X2))
                    | pp(aa(A,bool,Q,X2)) )
                <=> ( pp(aa(A,bool,P3,X2))
                    | pp(aa(A,bool,Q2,X2)) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_524_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X2))
         => ( X2 != T2 ) ) ) ).

% pinf(3)
tff(fact_525_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X2))
         => ( X2 != T2 ) ) ) ).

% pinf(4)
tff(fact_526_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),T2)) ) ) ).

% pinf(5)
tff(fact_527_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),X2)) ) ) ).

% pinf(7)
tff(fact_528_pinf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F4: D] :
        ? [Z3: C] :
        ! [X2: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),Z3),X2))
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_529_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [Z4: A] :
            ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Z4))
             => ( pp(aa(A,bool,P,X4))
              <=> pp(aa(A,bool,P3,X4)) ) )
         => ( ? [Z4: A] :
              ! [X4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Z4))
               => ( pp(aa(A,bool,Q,X4))
                <=> pp(aa(A,bool,Q2,X4)) ) )
           => ? [Z3: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z3))
               => ( ( pp(aa(A,bool,P,X2))
                    & pp(aa(A,bool,Q,X2)) )
                <=> ( pp(aa(A,bool,P3,X2))
                    & pp(aa(A,bool,Q2,X2)) ) ) ) ) ) ) ).

% minf(1)
tff(fact_530_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [Z4: A] :
            ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Z4))
             => ( pp(aa(A,bool,P,X4))
              <=> pp(aa(A,bool,P3,X4)) ) )
         => ( ? [Z4: A] :
              ! [X4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Z4))
               => ( pp(aa(A,bool,Q,X4))
                <=> pp(aa(A,bool,Q2,X4)) ) )
           => ? [Z3: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z3))
               => ( ( pp(aa(A,bool,P,X2))
                    | pp(aa(A,bool,Q,X2)) )
                <=> ( pp(aa(A,bool,P3,X2))
                    | pp(aa(A,bool,Q2,X2)) ) ) ) ) ) ) ).

% minf(2)
tff(fact_531_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z3))
         => ( X2 != T2 ) ) ) ).

% minf(3)
tff(fact_532_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z3))
         => ( X2 != T2 ) ) ) ).

% minf(4)
tff(fact_533_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),T2)) ) ) ).

% minf(5)
tff(fact_534_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),X2)) ) ) ).

% minf(7)
tff(fact_535_minf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F4: D] :
        ? [Z3: C] :
        ! [X2: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),X2),Z3))
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_536_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,N)))
     => ? [Info2: option(product_prod(nat,nat)),TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,N)),TreeList2,S4) ) ).

% deg_SUcn_Node
tff(fact_537_nth__zip,axiom,
    ! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys)))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I)) ) ) ) ).

% nth_zip
tff(fact_538_find__Some__iff2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),X) = find(A,P,Xs) )
    <=> ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4)))
          & ( X = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I4))
             => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_539_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),X) )
    <=> ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4)))
          & ( X = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I4))
             => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_540_count__notin,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(A,nat,count_list(A,Xs),X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_541__C4_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I3: nat] :
        ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m)))
       => ( ( ( vEBT_VEBT_high(ma,na) = I3 )
           => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I3)),vEBT_VEBT_low(ma,na))) )
          & ! [X2: nat] :
              ( ( ( vEBT_VEBT_high(X2,na) = I3 )
                & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I3)),vEBT_VEBT_low(X2,na))) )
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),mi),X2))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),ma)) ) ) ) ) ) ).

% "4.hyps"(9)
tff(fact_542_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2)))
    <=> ? [Y4: A,N3: nat] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N3)),Y4)),R2))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( Ys = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N3),Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_543_vebt__pred_Osimps_I6_J,axiom,
    ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(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)),Vh,Vi),Vj) = none(nat) ).

% vebt_pred.simps(6)
tff(fact_544_vebt__succ_Osimps_I5_J,axiom,
    ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(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)),Vg,Vh),Vi) = none(nat) ).

% vebt_succ.simps(5)
tff(fact_545_pair__lessI2,axiom,
    ! [A2: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
       => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_less)) ) ) ).

% pair_lessI2
tff(fact_546_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2)
     => ~ ! [N2: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) ) ).

% even_odd_cases
tff(fact_547_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_power,aa(nat,option(nat),some(nat),X)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% power_shift
tff(fact_548__C4_Ohyps_C_I8_J,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),deg))) ).

% "4.hyps"(8)
tff(fact_549__C4_Ohyps_C_I2_J,axiom,
    aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m) ).

% "4.hyps"(2)
tff(fact_550_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_551_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_552_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_553_dp,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),deg)) ).

% dp
tff(fact_554__C4_Ohyps_C_I5_J,axiom,
    ! [I3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m)))
     => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I3)),X_13))
      <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,summary),I3)) ) ) ).

% "4.hyps"(5)
tff(fact_555_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Ma,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))) ) ).

% high_bound_aux
tff(fact_556_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(Tree),X))
     => ( vEBT_invar_vebt(Tree,N)
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ) ).

% member_bound
tff(fact_557_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).

% numeral_le_iff
tff(fact_558_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).

% numeral_less_iff
tff(fact_559_lessI,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,N))) ).

% lessI
tff(fact_560_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N))) ) ).

% Suc_mono
tff(fact_561_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_eq
tff(fact_562_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(nat,nat,suc,M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% Suc_le_mono
tff(fact_563_misiz,axiom,
    ! [T2: vEBT_VEBT,N: nat,M: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( aa(nat,option(nat),some(nat),M) = vEBT_vebt_mint(T2) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ) ).

% misiz
tff(fact_564_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc_right
tff(fact_565_helpyd,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_succ(T2,X) = aa(nat,option(nat),some(nat),Y) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ) ).

% helpyd
tff(fact_566_helpypredd,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_pred(T2,X) = aa(nat,option(nat),some(nat),Y) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ) ).

% helpypredd
tff(fact_567__092_060open_062x_A_060_A2_A_094_Adeg_092_060close_062,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),deg))) ).

% \<open>x < 2 ^ deg\<close>
tff(fact_568__092_060open_062ma_A_092_060le_062_A2_A_094_Adeg_092_060close_062,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),deg))) ).

% \<open>ma \<le> 2 ^ deg\<close>
tff(fact_569_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_570_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) ) ) ) ).

% del_single_cont
tff(fact_571_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_572_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = aa(nat,option(nat),some(nat),Ma) ) ) ) ).

% pred_max
tff(fact_573_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = aa(nat,option(nat),some(nat),Mi) ) ) ) ).

% succ_min
tff(fact_574_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_575_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N))) ).

% zero_less_Suc
tff(fact_576_nothprolist,axiom,
    ! [I: nat] :
      ( ( ( I != vEBT_VEBT_high(xa,na) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m))) )
     => ( aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),I) = aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I) ) ) ).

% nothprolist
tff(fact_577_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Z))),fun_pair_less))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),Z)) ) ).

% pair_less_iff1
tff(fact_578_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_579_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_580_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_581_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_582_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_583_numeral__2__eq__2,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% numeral_2_eq_2
tff(fact_584_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_585_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_low(X,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_586_num_Osize_I5_J,axiom,
    ! [X22: num] : aa(num,nat,size_size(num),aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_587_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),N) ) ).

% zero_neq_numeral
tff(fact_588_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ( ( X != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va2: nat] : X != aa(nat,nat,suc,aa(nat,nat,suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_589_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_590_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_591_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_592_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_593_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_594_nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% nat_induct
tff(fact_595_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [X4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X4),zero_zero(nat)))
     => ( ! [Y2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,zero_zero(nat)),aa(nat,nat,suc,Y2)))
       => ( ! [X4: nat,Y2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X4),Y2))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,aa(nat,nat,suc,X4)),aa(nat,nat,suc,Y2))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ) ).

% diff_induct
tff(fact_596_zero__induct,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ).

% zero_induct
tff(fact_597_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_598_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_599_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_600_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M5: nat] : N = aa(nat,nat,suc,M5) ) ).

% not0_implies_Suc
tff(fact_601_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K))
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_602_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_lessD
tff(fact_603_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),K))
     => ~ ! [J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_604_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( ( aa(nat,nat,suc,M) != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_605_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_606_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% less_SucI
tff(fact_607_Ex__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,N))
        | ? [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
            & pp(aa(nat,bool,P,I4)) ) ) ) ).

% Ex_less_Suc
tff(fact_608_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_609_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M))) ) ).

% not_less_eq
tff(fact_610_All__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,N))
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
           => pp(aa(nat,bool,P,I4)) ) ) ) ).

% All_less_Suc
tff(fact_611_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
    <=> ? [M6: nat] :
          ( ( M = aa(nat,nat,suc,M6) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M6)) ) ) ).

% Suc_less_eq2
tff(fact_612_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_613_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_SucD
tff(fact_614_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_615_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( ! [I2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),aa(nat,nat,suc,I2)))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),J2))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,J2),K2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),K2)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_616_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => pp(aa(nat,bool,P,I2)) )
       => ( ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
             => ( pp(aa(nat,bool,P,aa(nat,nat,suc,I2)))
               => pp(aa(nat,bool,P,I2)) ) )
         => pp(aa(nat,bool,P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_617_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_618_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% Suc_leD
tff(fact_619_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_620_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N))) ) ).

% le_SucI
tff(fact_621_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M7))
     => ? [M5: nat] : M7 = aa(nat,nat,suc,M5) ) ).

% Suc_le_D
tff(fact_622_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        | ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_623_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_624_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_625_full__nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N2))
             => pp(aa(nat,bool,P,M2)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% full_nat_induct
tff(fact_626_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,P,M))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_627_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ! [X4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X4),X4))
       => ( ! [X4: nat,Y2: nat,Z3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X4),Y2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R,Y2),Z3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X4),Z3)) ) )
         => ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R,N2),aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_628_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_629_add__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc
tff(fact_630_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A2) )
     => ( aa(nat,nat,suc,A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_631_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_632_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_633_listrel1__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel1(A,S))) ) ).

% listrel1_mono
tff(fact_634_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X4,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = aa(nat,nat,suc,N) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_635_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X4,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_1))
               => ( ! [X4: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_1)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_636_find__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ( Xs = Ys )
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ys)))
           => ( pp(aa(A,bool,P,X4))
            <=> pp(aa(A,bool,Q,X4)) ) )
       => ( find(A,P,Xs) = find(A,Q,Ys) ) ) ) ).

% find_cong
tff(fact_637_zip__same,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs))))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Xs)))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_638_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),aa(list(B),set(B),set2(B),Ys))) ) ) ).

% in_set_zipE
tff(fact_639_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% set_zip_leftD
tff(fact_640_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),aa(list(B),set(B),set2(B),Ys))) ) ).

% set_zip_rightD
tff(fact_641_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,X: A,Ys: list(B),Y: B] : zip(A,B,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),I),Y)) = aa(product_prod(A,B),list(product_prod(A,B)),aa(nat,fun(product_prod(A,B),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(nat,fun(product_prod(A,B),list(product_prod(A,B)))),list_update(product_prod(A,B)),zip(A,B,Xs,Ys)),I),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_update
tff(fact_642_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_le_zero
tff(fact_643_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_le_numeral
tff(fact_644_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_less_numeral
tff(fact_645_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_less_zero
tff(fact_646_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N6: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,N6))) ) ) ) ).

% lift_Suc_mono_less
tff(fact_647_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,M: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,M)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_648_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N6: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,F2,N6))) ) ) ) ).

% lift_Suc_mono_le
tff(fact_649_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),N: nat,N6: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N2))),aa(nat,A,F2,N2)))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N6)),aa(nat,A,F2,N))) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_650_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        | ? [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,I4))) ) ) ) ).

% Ex_less_Suc2
tff(fact_651_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ? [M3: nat] : N = aa(nat,nat,suc,M3) ) ).

% gr0_conv_Suc
tff(fact_652_All__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,I4))) ) ) ) ).

% All_less_Suc2
tff(fact_653_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ? [M5: nat] : N = aa(nat,nat,suc,M5) ) ).

% gr0_implies_Suc
tff(fact_654_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_655_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% le_imp_less_Suc
tff(fact_656_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_657_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_Suc_eq_le
tff(fact_658_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_659_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_lessD
tff(fact_660_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,P,J))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),J))
               => ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
                 => pp(aa(nat,bool,P,N2)) ) ) )
         => pp(aa(nat,bool,P,I)) ) ) ) ).

% inc_induct
tff(fact_661_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,P,I))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),J))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) )
         => pp(aa(nat,bool,P,J)) ) ) ) ).

% dec_induct
tff(fact_662_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_eq
tff(fact_663_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N)) ) ).

% Suc_leI
tff(fact_664_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_665_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_666_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ? [K2: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_667_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_668_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)))) ).

% less_add_Suc2
tff(fact_669_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)))) ).

% less_add_Suc1
tff(fact_670_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ~ ! [Q3: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ) ).

% less_natE
tff(fact_671_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X4,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = aa(nat,nat,suc,N) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)))
                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_13))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma))
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I2 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X4: nat] :
                                    ( ( ( vEBT_VEBT_high(X4,N) = I2 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X4,N))) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X4))
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_672_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),aa(list(B),set(B),set2(B),Ys)))
       => ~ ! [X4: A] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_673_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ~ ! [Y2: B] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_674_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A2: A,B2: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,F2),aa(A,option(A),some(A),A2)),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F2,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_675_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uv: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uu),none(A)),Uv) = none(A) ).

% VEBT_internal.option_shift.simps(1)
tff(fact_676_ex__least__nat__less,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),N))
            & ! [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),K2))
               => ~ pp(aa(nat,bool,P,I3)) )
            & pp(aa(nat,bool,P,aa(nat,nat,suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_677_find__None__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( find(A,P,Xs) = none(A) )
    <=> ~ ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(A,bool,P,X3)) ) ) ).

% find_None_iff
tff(fact_678_find__None__iff2,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( none(A) = find(A,P,Xs) )
    <=> ~ ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(A,bool,P,X3)) ) ) ).

% find_None_iff2
tff(fact_679_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X4,N) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ! [I2: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)))
                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_13))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I2)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_1)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma))
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I2 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X4: nat] :
                                    ( ( ( vEBT_VEBT_high(X4,N) = I2 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X4,N))) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X4))
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_680_pair__lessI1,axiom,
    ! [A2: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_less)) ) ).

% pair_lessI1
tff(fact_681_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),X)),aa(list(A),nat,size_size(list(A)),Xs))) ).

% count_le_length
tff(fact_682_option_Osize_I4_J,axiom,
    ! [A: $tType,X22: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_683_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_684_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_685_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,A)),Xa2: option(A),Xb2: option(A),Y: option(A)] :
      ( ( aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,X),Xa2),Xb2) = Y )
     => ( ( ( Xa2 = none(A) )
         => ( Y != none(A) ) )
       => ( ( ? [V3: A] : Xa2 = aa(A,option(A),some(A),V3)
           => ( ( Xb2 = none(A) )
             => ( Y != none(A) ) ) )
         => ~ ! [A4: A] :
                ( ( Xa2 = aa(A,option(A),some(A),A4) )
               => ! [B4: A] :
                    ( ( Xb2 = aa(A,option(A),some(A),B4) )
                   => ( Y != aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),X,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_686_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw: fun(A,fun(A,A)),V2: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uw),aa(A,option(A),some(A),V2)),none(A)) = none(A) ).

% VEBT_internal.option_shift.simps(2)
tff(fact_687_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_688_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_689_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_690_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
            <=> ( X = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_691_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A)))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_692_post__member__pre__member,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
         => ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_vebt_insert(T2,X)),Y))
           => ( pp(aa(nat,bool,vEBT_vebt_member(T2),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_693_valid__insert__both__member__options__add,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
       => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T2,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_694_valid__insert__both__member__options__pres,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
         => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
           => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T2,Y)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_695_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
       => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_696_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ) ) ).

% power_mono_iff
tff(fact_697_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_698_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,N)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_699_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_700_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% power_Suc0_right
tff(fact_701_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_702_power__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_703_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_704_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% power_eq_0_iff
tff(fact_705_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,N: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_706_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono
tff(fact_707_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% zero_le_power
tff(fact_708_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% zero_less_power
tff(fact_709_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_710_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% power_less_imp_less_base
tff(fact_711_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_712_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% power_le_imp_le_base
tff(fact_713_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_714_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),K))) ) ).

% power_gt_expt
tff(fact_715_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N))) ) ).

% nat_one_le_power
tff(fact_716_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_717_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_718_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A2: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_719_less__exp,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% less_exp
tff(fact_720_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_721_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_722_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M))) ) ).

% self_le_ge2_pow
tff(fact_723_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% zero_le_power2
tff(fact_724_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_725_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% power2_le_imp_le
tff(fact_726_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A))) ) ).

% power2_less_0
tff(fact_727_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ) ).

% power_strict_mono
tff(fact_728_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% power2_less_imp_less
tff(fact_729_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_730_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% sum_power2_ge_zero
tff(fact_731_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A))) ) ).

% not_sum_power2_lt_zero
tff(fact_732_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_733_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
        & ( ( X = Mi )
          | ( X = Ma )
          | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
            & pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% member_inv
tff(fact_734_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_735_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),X))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_736_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg)))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
         => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_737_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),aa(num,num,bit0,N))) ).

% semiring_norm(76)
tff(fact_738_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X4,N) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N)) ) ) ).

% set_n_deg_not_0
tff(fact_739_vebt__insert_Osimps_I4_J,axiom,
    ! [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),X)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_740_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_741_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_742_bit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : vEBT_VEBT_bit_concat(H,L,D2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),D2))),L) ).

% bit_concat_def
tff(fact_743_pow__sum,axiom,
    ! [A2: nat,B2: nat] : divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2) ).

% pow_sum
tff(fact_744_high__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_high(X,N) = divide_divide(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% high_def
tff(fact_745__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = na ).

% \<open>deg div 2 = n\<close>
tff(fact_746_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_747_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_748_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_749_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_750_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_751_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),X),N) = Y ) ) ).

% high_inv
tff(fact_752_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),X),N) = X ) ) ).

% low_inv
tff(fact_753_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_754_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_755_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ).

% div_0
tff(fact_756_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( divide_divide(A,A2,B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_757_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_758_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( divide_divide(A,C2,A2) = divide_divide(A,C2,B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_759_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_760_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_761_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% mult_1
tff(fact_762_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.right_neutral
tff(fact_763_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_764_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_765_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_766_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_767_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_768_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_769_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(78)
tff(fact_770_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),one2)) ).

% semiring_norm(75)
tff(fact_771_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_772_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_773_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_774_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_775_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_776_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( C2 = zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A2,B2) ) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_777_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_778_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),A2) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_779_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_780_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_781_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),B2) = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_782_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_783_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( divide_divide(A,A2,B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_784_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_785_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = divide_divide(A,A2,B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_786_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_787_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( ( A2 = zero_zero(A) )
           => ( divide_divide(A,A2,A2) = zero_zero(A) ) )
          & ( ( A2 != zero_zero(A) )
           => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ) ).

% divide_self_if
tff(fact_788_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( divide_divide(A,B2,A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_789_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = divide_divide(A,B2,A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_790_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( divide_divide(A,one_one(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_791_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = divide_divide(A,one_one(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_792_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) )
          <=> ( M = N ) ) ) ) ).

% power_inject_exp
tff(fact_793_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_794_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_795_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_796_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% mult_less_cancel2
tff(fact_797_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_798_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc_right
tff(fact_799_less__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),one_one(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_800_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero(nat) )
       => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = zero_zero(nat) ) )
      & ( ( K != zero_zero(nat) )
       => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = divide_divide(nat,M,N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_801_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),Deg))
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_802_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% zero_le_divide_1_iff
tff(fact_803_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% divide_le_0_1_iff
tff(fact_804_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,one_one(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% divide_less_0_1_iff
tff(fact_805_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_806_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_807_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_808_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_809_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% zero_less_divide_1_iff
tff(fact_810_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2))),B2)) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_811_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_812_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W2: num] :
          ( ( A2 = divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)) )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W2) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2)) = B2 ) )
            & ( ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_813_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A2: A] :
          ( ( divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)) = A2 )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W2) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2)) ) )
            & ( ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_814_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_815_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2))),B2)) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_816_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,B2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_817_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_818_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y)) ) ) ) ).

% power_strict_increasing_iff
tff(fact_819_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),Deg))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
         => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_820_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),N)) ) ) ).

% one_le_mult_iff
tff(fact_821_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_822_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% mult_le_cancel2
tff(fact_823_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_824_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_825_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_826_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_827_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_828_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) ) ) ) ).

% power_increasing_iff
tff(fact_829_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),one2)) ) ) ).

% numeral_le_one_iff
tff(fact_830_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),N)) ) ) ).

% one_less_numeral_iff
tff(fact_831_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_832_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ) ) ) ).

% power_decreasing_iff
tff(fact_833__092_060open_062y_A_061_A_Iif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_092_060close_062,axiom,
    ( ( ( xa = ma )
     => ( ya = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_high(xa,na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),vEBT_VEBT_high(xa,na))))) ) )
    & ( ( xa != ma )
     => ( ya = ma ) ) ) ).

% \<open>y = (if x = ma then high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList [high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n)) else ma)\<close>
tff(fact_834_ssms,axiom,
    pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),xa),ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_high(xa,na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),vEBT_VEBT_high(xa,na))))),ma))),deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na))),summary)),ya)) ).

% ssms
tff(fact_835_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( divide_divide(A,A2,B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_836_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N))) ) ) ) ).

% less_1_mult
tff(fact_837_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
     => ( ( N = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_838_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = divide_divide(A,B2,C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_839_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( divide_divide(A,B2,C2) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_840_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 )
           => ( A2 = divide_divide(A,B2,C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_841_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
           => ( divide_divide(A,B2,C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_842_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = divide_divide(A,B2,C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq
tff(fact_843_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( divide_divide(A,B2,C2) = A2 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq
tff(fact_844_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( divide_divide(A,X,Y) = divide_divide(A,W2,Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_845_nat__mult__1__right,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).

% nat_mult_1_right
tff(fact_846_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_847_nat__mult__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).

% nat_mult_1
tff(fact_848_mult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.left_commute
tff(fact_849_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.comm_neutral
tff(fact_850_mult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) ) ).

% mult.commute
tff(fact_851_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.assoc
tff(fact_852_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_853_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_854_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_855_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = divide_divide(nat,M,N) ) ) ).

% nat_mult_div_cancel1
tff(fact_856_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),A2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ) ) ) ).

% divide_less_eq
tff(fact_857_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq
tff(fact_858_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% neg_divide_less_eq
tff(fact_859_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_less_divide_eq
tff(fact_860_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_divide_less_eq
tff(fact_861_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% pos_less_divide_eq
tff(fact_862_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),Z)) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_863_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),divide_divide(A,X,Y))) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_864_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,C2,A2)),divide_divide(A,C2,B2))) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_865_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,C2,A2)),divide_divide(A,C2,B2))) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_866_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W2) = divide_divide(A,B2,C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_867_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( divide_divide(A,B2,C2) = aa(num,A,numeral_numeral(A),W2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_868_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Z)),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Z)),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_869_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),divide_divide(A,B2,Z)) = A2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),divide_divide(A,B2,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),A2),Z)),B2),Z) ) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_870_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,Y)),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),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_871_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,X: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,Y)),Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Y) ) ) ) ).

% add_frac_num
tff(fact_872_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,X,Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Y) ) ) ) ).

% add_num_frac
tff(fact_873_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,Y,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),Y),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_874_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,Z)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_875_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A2)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_876_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% less_divide_eq_1
tff(fact_877_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))))) ) ) ).

% less_half_sum
tff(fact_878_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2)) ) ) ).

% gt_half_sum
tff(fact_879_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),A2)) ) ) ) ).

% mult_left_le
tff(fact_880_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A))) ) ) ) ) ).

% mult_le_one
tff(fact_881_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X)) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_882_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X)) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_883_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))) ) ) ).

% power_less_power_Suc
tff(fact_884_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))) ) ) ).

% power_gt1_lemma
tff(fact_885_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) )
         => ( ( A2 != zero_zero(A) )
            & ( B2 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_886_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
         => ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_887_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_888_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_889_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_890_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_891_mult__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% mult_0
tff(fact_892_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),one_one(A))) ) ).

% le_numeral_extra(4)
tff(fact_893_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_894_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),one_one(A))) ) ).

% less_numeral_extra(4)
tff(fact_895_le__cube,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)))) ).

% le_cube
tff(fact_896_le__square,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_square
tff(fact_897_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L))) ) ) ).

% mult_le_mono
tff(fact_898_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ).

% mult_le_mono1
tff(fact_899_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ).

% mult_le_mono2
tff(fact_900_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% add_mult_distrib2
tff(fact_901_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% add_mult_distrib
tff(fact_902_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_903_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_904_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),A2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ) ) ) ).

% divide_le_eq
tff(fact_905_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq
tff(fact_906_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,C2,A2)),divide_divide(A,C2,B2))) ) ) ) ) ).

% divide_left_mono
tff(fact_907_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% neg_divide_le_eq
tff(fact_908_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_le_divide_eq
tff(fact_909_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_divide_le_eq
tff(fact_910_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% pos_le_divide_eq
tff(fact_911_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),Z)) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_912_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),divide_divide(A,X,Y))) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_913_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,C2,A2)),divide_divide(A,C2,B2))) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_914_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_915_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_916_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_917_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% le_divide_eq_1
tff(fact_918_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_919_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2)) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_920_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_921_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2)) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_922_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_923_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2)) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_924_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_925_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2)) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_926_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),one_one(A)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),X)),Y)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_mult_one_interval
tff(fact_927_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A2: A,Y: A,U: A,V2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A2)) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_928_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ).

% power_Suc_less
tff(fact_929_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_930_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_931_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A2: A,Y: A,U: A,V2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A2)) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_932_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),divide_divide(A,A2,C2))) ) ) ) ).

% divide_right_mono_neg
tff(fact_933_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_934_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_935_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_936_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_937_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_938_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))) ) ) ) ).

% divide_right_mono
tff(fact_939_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A2,B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_le_0_iff
tff(fact_940_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_neg_neg
tff(fact_941_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_neg_pos
tff(fact_942_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_pos_neg
tff(fact_943_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_pos_pos
tff(fact_944_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A2,B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_less_0_iff
tff(fact_945_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_946_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A2,B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_947_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))) ) ) ) ).

% divide_strict_right_mono
tff(fact_948_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_949_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_950_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_951_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_952_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_953_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_954_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_955_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ).

% split_mult_neg_le
tff(fact_956_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_le_0_iff
tff(fact_957_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_right_mono
tff(fact_958_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_right_mono_neg
tff(fact_959_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_left_mono
tff(fact_960_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_961_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_left_mono_neg
tff(fact_962_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ).

% split_mult_pos_le
tff(fact_963_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2))) ) ).

% zero_le_square
tff(fact_964_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_mono'
tff(fact_965_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_mono
tff(fact_966_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_neg_neg
tff(fact_967_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),zero_zero(A))) ) ).

% not_square_less_zero
tff(fact_968_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_less_0_iff
tff(fact_969_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_neg_pos
tff(fact_970_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg
tff(fact_971_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_pos_pos
tff(fact_972_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg2
tff(fact_973_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_974_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos
tff(fact_975_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos2
tff(fact_976_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_977_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_978_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_979_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% mult_strict_left_mono
tff(fact_980_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_981_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_982_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).

% mult_strict_right_mono
tff(fact_983_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_984_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_985_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A2: A,B2: A,C2: A,D2: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_986_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_987_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_988_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) ) ).

% not_one_le_zero
tff(fact_989_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_990_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one_class.zero_le_one
tff(fact_991_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% less_numeral_extra(1)
tff(fact_992_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one
tff(fact_993_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) ) ).

% not_one_less_zero
tff(fact_994_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ) ).

% mult_less_mono1
tff(fact_995_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ) ).

% mult_less_mono2
tff(fact_996_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_997_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% one_le_numeral
tff(fact_998_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A))) ) ).

% not_numeral_less_one
tff(fact_999_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc
tff(fact_1000_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A)))) ) ) ).

% add_mono1
tff(fact_1001_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)))) ) ).

% less_add_one
tff(fact_1002_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% one_le_power
tff(fact_1003_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_1004_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_1005_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).

% Suc_eq_plus1_left
tff(fact_1006_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1007_Suc__eq__plus1,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_1008_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,W2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Z)),divide_divide(A,Y,W2))) ) ) ) ) ) ).

% frac_le
tff(fact_1009_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Z)),divide_divide(A,Y,W2))) ) ) ) ) ) ).

% frac_less
tff(fact_1010_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W2),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Z)),divide_divide(A,Y,W2))) ) ) ) ) ) ).

% frac_less2
tff(fact_1011_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% divide_le_cancel
tff(fact_1012_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_neg
tff(fact_1013_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonneg_pos
tff(fact_1014_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonpos_neg
tff(fact_1015_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_pos
tff(fact_1016_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1017_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1018_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1019_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1020_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1021_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1022_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1023_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1024_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1025_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1026_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1027_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1028_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1029_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1030_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))) ) ).

% sum_squares_ge_zero
tff(fact_1031_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1032_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))) ) ).

% not_sum_squares_lt_zero
tff(fact_1033_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1034_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% one_less_mult
tff(fact_1035_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_1036_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_1037_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ).

% zero_less_two
tff(fact_1038_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),one_one(A))) ) ) ) ).

% power_le_one
tff(fact_1039_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N)))) ) ) ).

% power_gt1
tff(fact_1040_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ) ).

% power_0_left
tff(fact_1041_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1042_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N4))) ) ) ) ).

% power_strict_increasing
tff(fact_1043_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N4))) ) ) ) ).

% power_increasing
tff(fact_1044_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_1045_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N))),A2)) ) ) ) ).

% power_Suc_le_self
tff(fact_1046_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).

% power_Suc_less_one
tff(fact_1047_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1048_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N4: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ) ).

% power_decreasing
tff(fact_1049_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1050_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ).

% self_le_power
tff(fact_1051_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ) ).

% one_less_power
tff(fact_1052_nat__bit__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => pp(aa(nat,bool,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_1053_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ).

% half_gt_zero
tff(fact_1054_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% half_gt_zero_iff
tff(fact_1055_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ).

% field_less_half_sum
tff(fact_1056_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ) ).

% zero_le_even_power'
tff(fact_1057_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1058_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),zero_zero(A))) ) ) ).

% odd_power_less_zero
tff(fact_1059_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
       => ? [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl2
tff(fact_1060_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
       => ? [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl1
tff(fact_1061_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts,S),X) = vEBT_Node(Info,zero_zero(nat),Ts,S) ).

% vebt_insert.simps(2)
tff(fact_1062_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),X) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_1063_aaaa,axiom,
    ( ( ya = mi )
    | ( ( ( xa = ma )
       => ( ya = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_high(xa,na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),vEBT_VEBT_high(xa,na))))) ) )
      & ( ( xa != ma )
       => ( ya = ma ) ) )
    | ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),vEBT_VEBT_high(ya,na))),vEBT_VEBT_low(ya,na)))
      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(ya,na)),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))))) ) ) ).

% aaaa
tff(fact_1064__092_060open_062vebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Imi_M_Aif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_J_Adeg_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_J_Asummary_092_060close_062,axiom,
    vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary),xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),xa),ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_high(xa,na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),vEBT_VEBT_high(xa,na))))),ma))),deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na))),summary) ).

% \<open>vebt_delete (Node (Some (mi, ma)) deg treeList summary) x = Node (Some (mi, if x = ma then high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList [high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n)) else ma)) deg (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)]) summary\<close>
tff(fact_1065_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_1066__092_060open_062_Iif_Ax_A_061_Ama_Athen_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_Aelse_Ama_J_A_061_Ahigh_Ax_An_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__maxt_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Ahigh_Ax_An_J_J_092_060close_062,axiom,
    ( ( xa != ma )
   => ( ma = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_high(xa,na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),treeList),vEBT_VEBT_high(xa,na)),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),vEBT_VEBT_low(xa,na)))),vEBT_VEBT_high(xa,na))))) ) ) ).

% \<open>(if x = ma then high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList [high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n)) else ma) = high x n * 2 ^ (deg div 2) + the (vebt_maxt (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)] ! high x n))\<close>
tff(fact_1067_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = L )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),Newnode) )
                     => ( ~ pp(vEBT_VEBT_minNull(Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_1068_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = H )
           => ( ( vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ~ pp(vEBT_VEBT_minNull(Newnode))
                 => ( ( Newlist = aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),Newnode) )
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_1069_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list(vEBT_VEBT),X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( X != 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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mi),Ma))),Deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(vEBT_VEBT,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),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_1070_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( X != 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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Ma))),Deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_1071_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),N)
     => ( ( N = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),Mi))
         => ( ( Ma != Mi )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Va,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),aa(nat,nat,suc,divide_divide(nat,Va,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ).

% nested_mint
tff(fact_1072_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_1073_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% half_negative_int_iff
tff(fact_1074_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),bot_bot(A)) = X ) ).

% max_bot2
tff(fact_1075_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),X) = X ) ).

% max_bot
tff(fact_1076_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)) ).

% max_Suc_Suc
tff(fact_1077_max__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),zero_zero(nat)) = N ).

% max_0R
tff(fact_1078_max__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N) = N ).

% max_0L
tff(fact_1079_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_1080_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1081_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_1082_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_1083_summaxma,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),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi != Ma )
       => ( aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Summary)) = vEBT_VEBT_high(Ma,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% summaxma
tff(fact_1084_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( ( C2 = zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = zero_zero(A) ) )
          & ( ( C2 != zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A2,B2) ) ) ) ) ).

% div_mult_mult1_if
tff(fact_1085_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult2
tff(fact_1086_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult1
tff(fact_1087_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(1)
tff(fact_1088_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(3)
tff(fact_1089_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(4)
tff(fact_1090_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_1091_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_1092_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_1093_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( divide_divide(nat,M,N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_1094_option_Ocollapse,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) = Option ) ) ).

% option.collapse
tff(fact_1095_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self4
tff(fact_1096_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self3
tff(fact_1097_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self2
tff(fact_1098_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self1
tff(fact_1099_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1100_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_1101_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_1102_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ) ).

% max_def
tff(fact_1103_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_1104_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% max_add_distrib_right
tff(fact_1105_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q4)) ).

% nat_add_max_left
tff(fact_1106_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q4)) ).

% nat_add_max_right
tff(fact_1107_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)) ).

% nat_mult_max_left
tff(fact_1108_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)) ).

% nat_mult_max_right
tff(fact_1109_option_Osel,axiom,
    ! [A: $tType,X22: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X22)) = X22 ).

% option.sel
tff(fact_1110_option_Oexpand,axiom,
    ! [A: $tType,Option: option(A),Option2: option(A)] :
      ( ( ( Option = none(A) )
      <=> ( Option2 = none(A) ) )
     => ( ( ( Option != none(A) )
         => ( ( Option2 != none(A) )
           => ( aa(option(A),A,the2(A),Option) = aa(option(A),A,the2(A),Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_1111_option_Oexhaust__sel,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) ) ) ).

% option.exhaust_sel
tff(fact_1112_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,M,N)),M)) ).

% div_le_dividend
tff(fact_1113_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,M,K)),divide_divide(nat,N,K))) ) ).

% div_le_mono
tff(fact_1114_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide(nat,M,N) = zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1115_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,M,N)),divide_divide(nat,aa(nat,nat,suc,M),N))) ).

% Suc_div_le_mono
tff(fact_1116_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M,N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_1117_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_1118_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),divide_divide(nat,M,N))),M)) ).

% times_div_less_eq_dividend
tff(fact_1119_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1120_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1121_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),divide_divide(nat,M,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% div_greater_zero_iff
tff(fact_1122_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,K,N)),divide_divide(nat,K,M))) ) ) ).

% div_le_mono2
tff(fact_1123_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M,Q4)),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4))) ) ) ).

% div_less_iff_less_mult
tff(fact_1124_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( ( divide_divide(nat,M,N) = M )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1125_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M,N)),M)) ) ) ).

% div_less_dividend
tff(fact_1126_div__nat__eqI,axiom,
    ! [N: nat,Q4: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q4))))
       => ( divide_divide(nat,M,N) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_1127_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),divide_divide(nat,N,Q4)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1128_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),divide_divide(nat,M,N))))) ) ).

% dividend_less_times_div
tff(fact_1129_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,N)),N)))) ) ).

% dividend_less_div_times
tff(fact_1130_split__div,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,divide_divide(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ( ( N != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I4)),J3) )
               => pp(aa(nat,bool,P,I4)) ) ) ) ) ) ).

% split_div
tff(fact_1131_split__div_H,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,divide_divide(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
          & pp(aa(nat,bool,P,zero_zero(nat))) )
        | ? [Q5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)),M))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q5))))
            & pp(aa(nat,bool,P,Q5)) ) ) ) ).

% split_div'
tff(fact_1132_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% div_2_gt_zero
tff(fact_1133_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),divide_divide(nat,aa(nat,nat,suc,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1134_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,X: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ) ) ).

% arith_geo_mean
tff(fact_1135_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),N)
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),Ma))
          & ? [M5: nat] :
              ( ( aa(nat,option(nat),some(nat),M5) = vEBT_vebt_mint(Summary) )
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ) ) ) ).

% mintlistlength
tff(fact_1136_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% sum_squares_bound
tff(fact_1137_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_1138_nat__induct2,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct2
tff(fact_1139_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% set_bit_0
tff(fact_1140_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A7: bool,B7: bool] : A1 = vEBT_Leaf(A7,B7)
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N3: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X3,N3) )
            & vEBT_invar_vebt(Summary2,N3)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N3) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),N3) )
            & ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_13))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N3: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X3,N3) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N3))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N3)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(nat,nat,suc,N3)) )
            & ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_13))
            & ! [X3: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),A22,TreeList3,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X3,N3) )
            & vEBT_invar_vebt(Summary2,N3)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N3) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),N3) )
            & ! [I4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N3)))
               => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),X_13))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_13)) ) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi2),Ma2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22)))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N3)))
                 => ( ( ( vEBT_VEBT_high(Ma2,N3) = I4 )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(Ma2,N3))) )
                    & ! [X3: nat] :
                        ( ( ( vEBT_VEBT_high(X3,N3) = I4 )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(X3,N3))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X3))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Ma2)) ) ) ) ) ) )
        | ? [TreeList3: list(vEBT_VEBT),N3: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),A22,TreeList3,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
               => vEBT_invar_vebt(X3,N3) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N3))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N3)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(nat,nat,suc,N3)) )
            & ! [I4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N3))))
               => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),X_13))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                 => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_13)) ) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi2),Ma2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22)))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N3))))
                 => ( ( ( vEBT_VEBT_high(Ma2,N3) = I4 )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(Ma2,N3))) )
                    & ! [X3: nat] :
                        ( ( ( vEBT_VEBT_high(X3,N3) = I4 )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(X3,N3))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X3))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Ma2)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1141_Leaf__0__not,axiom,
    ! [A2: bool,B2: bool] : ~ vEBT_invar_vebt(vEBT_Leaf(A2,B2),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_1142_deg__1__Leafy,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( N = one_one(nat) )
       => ? [A4: bool,B4: bool] : T2 = vEBT_Leaf(A4,B4) ) ) ).

% deg_1_Leafy
tff(fact_1143_deg__1__Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,one_one(nat))
     => ? [A4: bool,B4: bool] : T2 = vEBT_Leaf(A4,B4) ) ).

% deg_1_Leaf
tff(fact_1144_deg1Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,one_one(nat))
    <=> ? [A7: bool,B7: bool] : T2 = vEBT_Leaf(A7,B7) ) ).

% deg1Leaf
tff(fact_1145_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),X)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% mul_shift
tff(fact_1146_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z)) ) ).

% zle_add1_eq_le
tff(fact_1147_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X222: bool,Y21: bool,Y222: bool] :
      ( ( vEBT_Leaf(X21,X222) = vEBT_Leaf(Y21,Y222) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X222)
        <=> pp(Y222) ) ) ) ).

% VEBT.inject(2)
tff(fact_1148_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),A2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)) ) ) ).

% power_minus_is_div
tff(fact_1149_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% diff_self
tff(fact_1150_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_0_right
tff(fact_1151_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_diff
tff(fact_1152_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_zero
tff(fact_1153_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1154_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel_right'
tff(fact_1155_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_right
tff(fact_1156_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),A2) = B2 ) ).

% add_diff_cancel_left'
tff(fact_1157_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_left
tff(fact_1158_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_1159_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel
tff(fact_1160_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),K) ).

% Suc_diff_diff
tff(fact_1161_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% diff_Suc_Suc
tff(fact_1162_diff__0__eq__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_1163_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1164_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1165_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_1166_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5668285175392031749et_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% set_bit_negative_int_iff
tff(fact_1167_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1168_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1169_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1170_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_1171_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_1172_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_1173_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% zero_less_diff
tff(fact_1174_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_1175_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% diff_is_0_eq
tff(fact_1176_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1177_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1178_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1179_diff__Suc__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),one_one(nat)) = N ).

% diff_Suc_1
tff(fact_1180_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1181_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1182_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% Suc_pred
tff(fact_1183_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1184_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1185_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = N ) ) ).

% Suc_diff_1
tff(fact_1186_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1187_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_1188_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_1189_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z)) ) ).

% add1_zle_eq
tff(fact_1190_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z))) ) ).

% le_imp_0_less
tff(fact_1191_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z)) ) ).

% zless_imp_add1_zle
tff(fact_1192_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X4: int] :
            ( pp(aa(int,bool,P,X4))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D2))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X2: int] :
              ( pp(aa(int,bool,P,X2))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1193_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_gr_induct
tff(fact_1194_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z))
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_1195_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ).

% odd_less_0_iff
tff(fact_1196_less__int__code_I1_J,axiom,
    ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),zero_zero(int))) ).

% less_int_code(1)
tff(fact_1197_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),K)),J) ).

% diff_commute
tff(fact_1198_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) ) ).

% diff_right_commute
tff(fact_1199_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( ( A2 = B2 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_1200_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf(X21,X222)) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_1201_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2))) ) ) ) ).

% diff_mono
tff(fact_1202_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))) ) ) ).

% diff_left_mono
tff(fact_1203_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).

% diff_right_mono
tff(fact_1204_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2)) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1205_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_1206_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: bool,X222: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X222) ).

% VEBT.distinct(1)
tff(fact_1207_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: bool,X223: bool] : Y != vEBT_Leaf(X212,X223) ) ).

% VEBT.exhaust
tff(fact_1208_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,D3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),D3)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,Deg3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary3)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_1209_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),D2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2))) ) ) ) ).

% diff_strict_mono
tff(fact_1210_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2)) ) ) ) ).

% diff_eq_diff_less
tff(fact_1211_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))) ) ) ).

% diff_strict_left_mono
tff(fact_1212_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).

% diff_strict_right_mono
tff(fact_1213_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_1214_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A2 )
         => ( C2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_1215_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1216_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_1217_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_1218_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_1219_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_1220_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = C2 )
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_1221_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_1222_zero__induct__lemma,axiom,
    ! [P: fun(nat,bool),K: nat,I: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I))) ) ) ).

% zero_induct_lemma
tff(fact_1223_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_1224_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) = zero_zero(nat) )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_1225_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ) ).

% diff_less_mono2
tff(fact_1226_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_1227_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_1228_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% le_diff_iff
tff(fact_1229_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1230_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),L))) ) ).

% diff_le_mono
tff(fact_1231_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ).

% diff_le_self
tff(fact_1232_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),C2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),C2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),A2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),B2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),A2)) ) ) ) ).

% le_diff_iff'
tff(fact_1233_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ).

% diff_le_mono2
tff(fact_1234_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = M ).

% diff_add_inverse2
tff(fact_1235_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),N) = M ).

% diff_add_inverse
tff(fact_1236_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% diff_cancel2
tff(fact_1237_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% Nat.diff_cancel
tff(fact_1238_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% diff_mult_distrib2
tff(fact_1239_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% diff_mult_distrib
tff(fact_1240_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_1241_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(vEBT_VEBT_minNull(vEBT_Leaf(Uu,fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1242_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(vEBT_VEBT_minNull(vEBT_Leaf(fTrue,Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1243_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(vEBT_VEBT_minNull(vEBT_Leaf(fFalse,fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1244_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_1245_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A))) ) ) ).

% le_iff_diff_le_0
tff(fact_1246_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A))) ) ) ).

% less_iff_diff_less_0
tff(fact_1247_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_le_eq
tff(fact_1248_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2)) ) ) ).

% le_diff_eq
tff(fact_1249_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),A2) = B2 ) ) ) ).

% diff_add
tff(fact_1250_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2))) ) ) ).

% le_add_diff
tff(fact_1251_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1252_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1253_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1254_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1255_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1256_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1257_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1258_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1259_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A,J: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K)),J)) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1260_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K))) ) ) ).

% add_le_imp_le_diff
tff(fact_1261_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2)) ) ) ).

% less_diff_eq
tff(fact_1262_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_less_eq
tff(fact_1263_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1264_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: bool,B2: bool,N: nat] : vEBT_vebt_delete(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,N))) = vEBT_Leaf(A2,B2) ).

% vebt_delete.simps(3)
tff(fact_1265_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_vebt_delete(vEBT_Leaf(A2,B2),zero_zero(nat)) = vEBT_Leaf(fFalse,B2) ).

% vebt_delete.simps(1)
tff(fact_1266_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),aa(nat,nat,suc,M))) ).

% diff_less_Suc
tff(fact_1267_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ) ) ).

% Suc_diff_Suc
tff(fact_1268_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ) ) ).

% diff_less
tff(fact_1269_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) ) ).

% Suc_diff_le
tff(fact_1270_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),A2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C2))) ) ) ).

% diff_less_mono
tff(fact_1271_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% less_diff_iff
tff(fact_1272_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1273_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1274_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ).

% less_diff_conv
tff(fact_1275_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ).

% le_diff_conv
tff(fact_1276_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_1277_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1278_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1279_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1280_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1281_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_1282_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D2: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv),D2)
    <=> ( D2 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_1283_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),M) ).

% nat_minus_add_max
tff(fact_1284_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: bool,B4: bool,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),X4)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Ux2)
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4)),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_1285_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E3)),D2))) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1286_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E3)),C2)),D2)) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1287_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E3)),D2))) ) ) ).

% less_add_iff2
tff(fact_1288_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E3)),C2)),D2)) ) ) ).

% less_add_iff1
tff(fact_1289_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_invar_vebt(vEBT_Leaf(A2,B2),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_1290_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,X,Z)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1291_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,Y,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),Y),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1292_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,X,Y)),divide_divide(A,W2,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1293_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),divide_divide(A,B2,Z)) = A2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),divide_divide(A,B2,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),Z)),B2),Z) ) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1294_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_vebt_delete(vEBT_Leaf(A2,B2),aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf(A2,fFalse) ).

% vebt_delete.simps(2)
tff(fact_1295_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_1296_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_1297_vebt__member_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Leaf(A2,B2)),X))
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A2) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_1298_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_1299_nat__diff__split,axiom,
    ! [P: fun(nat,bool),A2: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ! [D5: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
           => pp(aa(nat,bool,P,D5)) ) ) ) ).

% nat_diff_split
tff(fact_1300_nat__diff__split__asm,axiom,
    ! [P: fun(nat,bool),A2: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)))
    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
            & ~ pp(aa(nat,bool,P,zero_zero(nat))) )
          | ? [D5: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
              & ~ pp(aa(nat,bool,P,D5)) ) ) ) ).

% nat_diff_split_asm
tff(fact_1301_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ) ).

% less_diff_conv2
tff(fact_1302_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(vEBT_VEBT_minNull(X))
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1303_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A2: bool,B2: bool] :
      ( ( ( X = zero_zero(nat) )
       => ( vEBT_vebt_insert(vEBT_Leaf(A2,B2),X) = vEBT_Leaf(fTrue,B2) ) )
      & ( ( X != zero_zero(nat) )
       => ( ( ( X = one_one(nat) )
           => ( vEBT_vebt_insert(vEBT_Leaf(A2,B2),X) = vEBT_Leaf(A2,fTrue) ) )
          & ( ( X != one_one(nat) )
           => ( vEBT_vebt_insert(vEBT_Leaf(A2,B2),X) = vEBT_Leaf(A2,B2) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_1304_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1305_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1306_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_1307_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_1308_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1309_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1310_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf(A2,B2),X)
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A2) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_1311_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw: bool,N: nat] : vEBT_vebt_succ(vEBT_Leaf(Uv,Uw),aa(nat,nat,suc,N)) = none(nat) ).

% vebt_succ.simps(2)
tff(fact_1312_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : vEBT_vebt_pred(vEBT_Leaf(Uu,Uv),zero_zero(nat)) = none(nat) ).

% vebt_pred.simps(1)
tff(fact_1313_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(vEBT_VEBT_minNull(X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1314_realpow__pos__nth2,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ? [R3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),aa(nat,nat,suc,N)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1315_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),divide_divide(A,W2,Z)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A))) ) ) ) ) ).

% frac_le_eq
tff(fact_1316_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),divide_divide(A,W2,Z)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A))) ) ) ) ) ).

% frac_less_eq
tff(fact_1317_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ) ) ) ).

% power_diff
tff(fact_1318_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A4: bool,B4: bool] : X != vEBT_Leaf(A4,B4)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
       => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),Ux2,Uy2,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_1319_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
          | ( N = zero_zero(nat) ) )
       => ( divide_divide(nat,M,N) = zero_zero(nat) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
            | ( N = zero_zero(nat) ) )
       => ( divide_divide(nat,M,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N)) ) ) ) ).

% div_if
tff(fact_1320_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_1321_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1322_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).

% add_eq_if
tff(fact_1323_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_1324_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_1325_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).

% mult_eq_if
tff(fact_1326_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(vEBT_VEBT_minNull(X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1327_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R2: A,S: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R2),S))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,aa(A,fun(A,A),minus_minus(A),V2),U)),S))),V2)) ) ) ) ) ).

% scaling_mono
tff(fact_1328_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,M: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1329_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
             => ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) )
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
             => ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) = divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ) ) ).

% power_diff_power_eq
tff(fact_1330_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [M: nat,P2: A] :
          ( ( ( M = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M) = aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ) ).

% power_eq_if
tff(fact_1331_vebt__maxt_Osimps_I1_J,axiom,
    ! [B2: bool,A2: bool] :
      ( ( pp(B2)
       => ( vEBT_vebt_maxt(vEBT_Leaf(A2,B2)) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
      & ( ~ pp(B2)
       => ( ( pp(A2)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A2,B2)) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
          & ( ~ pp(A2)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A2,B2)) = none(nat) ) ) ) ) ) ).

% vebt_maxt.simps(1)
tff(fact_1332_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),A2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N) ) ) ) ).

% power_minus_mult
tff(fact_1333_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool] :
      ( ( pp(A2)
       => ( vEBT_vebt_mint(vEBT_Leaf(A2,B2)) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
      & ( ~ pp(A2)
       => ( ( pp(B2)
           => ( vEBT_vebt_mint(vEBT_Leaf(A2,B2)) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
          & ( ~ pp(B2)
           => ( vEBT_vebt_mint(vEBT_Leaf(A2,B2)) = none(nat) ) ) ) ) ) ).

% vebt_mint.simps(1)
tff(fact_1334_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N)))) ) ).

% diff_le_diff_pow
tff(fact_1335_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ( divide_divide(nat,M,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N)) ) ) ) ).

% le_div_geq
tff(fact_1336_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: bool,Uw: bool] :
      ( ( pp(A2)
       => ( vEBT_vebt_pred(vEBT_Leaf(A2,Uw),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
      & ( ~ pp(A2)
       => ( vEBT_vebt_pred(vEBT_Leaf(A2,Uw),aa(nat,nat,suc,zero_zero(nat))) = none(nat) ) ) ) ).

% vebt_pred.simps(2)
tff(fact_1337_vebt__succ_Osimps_I1_J,axiom,
    ! [B2: bool,Uu: bool] :
      ( ( pp(B2)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B2),zero_zero(nat)) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
      & ( ~ pp(B2)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B2),zero_zero(nat)) = none(nat) ) ) ) ).

% vebt_succ.simps(1)
tff(fact_1338_vebt__insert_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: bool,B4: bool,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),X4)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S4)),X4)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4)),X4)
         => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3)),X4)
           => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),X4) ) ) ) ) ).

% vebt_insert.cases
tff(fact_1339_vebt__pred_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),zero_zero(nat))
     => ( ! [A4: bool,Uw2: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,Uw2)),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: bool,B4: bool,Va2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT,Vb2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Vb2)
           => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)),Vf2)
             => ( ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Vj2)
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),X4) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_1340_vebt__succ_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,B4: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,B4)),zero_zero(nat))
     => ( ! [Uv2: bool,Uw2: bool,N2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uv2,Uw2)),aa(nat,nat,suc,N2))
       => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Va3)
         => ( ! [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)),Ve2)
           => ( ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Vi2)
             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),X4) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_1341_vebt__delete_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: bool,B4: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),zero_zero(nat))
     => ( ! [A4: bool,B4: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: bool,B4: bool,N2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),aa(nat,nat,suc,aa(nat,nat,suc,N2)))
         => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,Uu2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3)),Uu2)
           => ( ! [Mi3: nat,Ma3: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),TrLst2,Smry2)),X4)
             => ( ! [Mi3: nat,Ma3: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2)),X4)
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),X4) ) ) ) ) ) ) ).

% vebt_delete.cases
tff(fact_1342_vebt__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: bool,B4: bool,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),X4)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),X4)
       => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),X4)
         => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X4)
           => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),X4) ) ) ) ) ).

% vebt_member.cases
tff(fact_1343_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Uw2)
     => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Uz2)
       => ( ! [Mi3: nat,Ma3: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2)),X4)
         => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2)),X4)
           => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT,X4: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1344_vebt__pred_Osimps_I3_J,axiom,
    ! [B2: bool,A2: bool,Va: nat] :
      ( ( pp(B2)
       => ( vEBT_vebt_pred(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
      & ( ~ pp(B2)
       => ( ( pp(A2)
           => ( vEBT_vebt_pred(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
          & ( ~ pp(A2)
           => ( vEBT_vebt_pred(vEBT_Leaf(A2,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = none(nat) ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_1345_realpow__pos__nth,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => ? [R3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),N) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1346_realpow__pos__nth__unique,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => ? [X4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X4))
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X4),N) = A2 )
            & ! [Y3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3))
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y3),N) = A2 ) )
               => ( Y3 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1347_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( pp(B4)
                 => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                & ( ~ pp(B4)
                 => ( ( pp(A4)
                     => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                    & ( ~ pp(A4)
                     => ( Y = none(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi3: nat,Ma3: nat] :
                ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma3) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_1348_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( pp(A4)
                 => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                & ( ~ pp(A4)
                 => ( ( pp(B4)
                     => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                    & ( ~ pp(B4)
                     => ( Y = none(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi3: nat] :
                ( ? [Ma3: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi3) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_1349_pos2,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% pos2
tff(fact_1350_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: bool,B4: bool] : A1 = vEBT_Leaf(A4,B4)
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3) )
             => ( ( A22 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                     => vEBT_invar_vebt(X2,N2) )
                 => ( vEBT_invar_vebt(Summary3,M5)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5) )
                     => ( ( M5 = N2 )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M5) )
                         => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_12))
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                 => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3) )
               => ( ( A22 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_invar_vebt(X2,N2) )
                   => ( vEBT_invar_vebt(Summary3,M5)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5) )
                       => ( ( M5 = aa(nat,nat,suc,N2) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M5) )
                           => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_12))
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                   => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),Deg2,TreeList2,Summary3) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_invar_vebt(X2,N2) )
                     => ( vEBT_invar_vebt(Summary3,M5)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5) )
                         => ( ( M5 = N2 )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M5) )
                             => ( ! [I3: nat] :
                                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5)))
                                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_13))
                                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
                               => ( ( ( Mi3 = Ma3 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                       => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi3),Ma3))
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg2)))
                                     => ~ ( ( Mi3 != Ma3 )
                                         => ! [I3: nat] :
                                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5)))
                                             => ( ( ( vEBT_VEBT_high(Ma3,N2) = I3 )
                                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma3,N2))) )
                                                & ! [X2: nat] :
                                                    ( ( ( vEBT_VEBT_high(X2,N2) = I3 )
                                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X2,N2))) )
                                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X2))
                                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Ma3)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                    ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),Deg2,TreeList2,Summary3) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                           => vEBT_invar_vebt(X2,N2) )
                       => ( vEBT_invar_vebt(Summary3,M5)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5) )
                           => ( ( M5 = aa(nat,nat,suc,N2) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M5) )
                               => ( ! [I3: nat] :
                                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5)))
                                     => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_13))
                                      <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I3)) ) )
                                 => ( ( ( Mi3 = Ma3 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                                         => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi3),Ma3))
                                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg2)))
                                       => ~ ( ( Mi3 != Ma3 )
                                           => ! [I3: nat] :
                                                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5)))
                                               => ( ( ( vEBT_VEBT_high(Ma3,N2) = I3 )
                                                   => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma3,N2))) )
                                                  & ! [X2: nat] :
                                                      ( ( ( vEBT_VEBT_high(X2,N2) = I3 )
                                                        & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X2,N2))) )
                                                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X2))
                                                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Ma3)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_1351_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
       => ( divide_divide(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),K),M),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1352_inrange,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(T2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),one_one(nat))))) ) ).

% inrange
tff(fact_1353_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb3
tff(fact_1354_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_1355_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z)) ) ) ) ).

% max_less_iff_conj
tff(fact_1356_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_1357_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_1358_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% max.bounded_iff
tff(fact_1359_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( divide_divide(nat,M,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N)) ) ) ) ).

% div_geq
tff(fact_1360_Diff__cancel,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),A3) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1361_empty__Diff,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A3) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1362_Diff__empty,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),bot_bot(set(A))) = A3 ).

% Diff_empty
tff(fact_1363_finite__Diff2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)))
      <=> pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% finite_Diff2
tff(fact_1364_finite__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ) ).

% finite_Diff
tff(fact_1365_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% Diff_eq_empty_iff
tff(fact_1366_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or1337092689740270186AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),U)) ) ) ) ).

% atLeastAtMost_iff
tff(fact_1367_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 ) )
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
              & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L2),H2)) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_1368_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)))
    <=> ( X = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1369_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_or1337092689740270186AtMost(nat,L,U))) ).

% finite_atLeastAtMost
tff(fact_1370_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% atLeastatMost_empty_iff
tff(fact_1371_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_1372_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1373_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_1374_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or1337092689740270186AtMost(A,A2,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% infinite_Icc_iff
tff(fact_1375_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z)) ) ).

% zle_diff1_eq
tff(fact_1376_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N2))) ) ).

% real_arch_pow
tff(fact_1377_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N2)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_1378_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_1379_Diff__infinite__finite,axiom,
    ! [A: $tType,T3: set(A),S2: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),T3))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
       => ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T3))) ) ) ).

% Diff_infinite_finite
tff(fact_1380_double__diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1381_Diff__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),A3)) ).

% Diff_subset
tff(fact_1382_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),D6: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D6),B3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C3),D6))) ) ) ).

% Diff_mono
tff(fact_1383_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_less_induct
tff(fact_1384_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ? [B4: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))) ) ).

% psubset_imp_ex_mem
tff(fact_1385_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or1337092689740270186AtMost(A,A2,B2))) ) ) ).

% infinite_Icc
tff(fact_1386_ex__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N))
          & pp(aa(nat,bool,P,M3)) )
    <=> ? [X3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X3)) ) ) ).

% ex_nat_less
tff(fact_1387_all__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N))
         => pp(aa(nat,bool,P,M3)) )
    <=> ! [X3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X3)) ) ) ).

% all_nat_less
tff(fact_1388_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N4)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_1389_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X4: int] :
            ( pp(aa(int,bool,P,X4))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D2))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X2: int] :
              ( pp(aa(int,bool,P,X2))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1390_plusinfinity,axiom,
    ! [D2: int,P3: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X4: int,K2: int] :
            ( pp(aa(int,bool,P3,X4))
          <=> pp(aa(int,bool,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),X4))
             => ( pp(aa(int,bool,P,X4))
              <=> pp(aa(int,bool,P3,X4)) ) )
         => ( ? [X_12: int] : pp(aa(int,bool,P3,X_12))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_1391_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X4: int,K2: int] :
            ( pp(aa(int,bool,P1,X4))
          <=> pp(aa(int,bool,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X4),Z4))
             => ( pp(aa(int,bool,P,X4))
              <=> pp(aa(int,bool,P1,X4)) ) )
         => ( ? [X_12: int] : pp(aa(int,bool,P1,X_12))
           => ? [X_1: int] : pp(aa(int,bool,P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_1392_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
       => ( divide_divide(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1393_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
              | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
                & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2))
                  | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2)) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_1394_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ) ).

% max.mono
tff(fact_1395_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.orderE
tff(fact_1396_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% max.orderI
tff(fact_1397_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% max.boundedE
tff(fact_1398_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)) ) ) ) ).

% max.boundedI
tff(fact_1399_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.order_iff
tff(fact_1400_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ).

% max.cobounded1
tff(fact_1401_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ).

% max.cobounded2
tff(fact_1402_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y)) ) ) ) ).

% le_max_iff_disj
tff(fact_1403_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb_iff1
tff(fact_1404_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_1405_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ).

% max.coboundedI1
tff(fact_1406_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ).

% max.coboundedI2
tff(fact_1407_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),Y)) ) ) ) ).

% less_max_iff_disj
tff(fact_1408_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% max.strict_boundedE
tff(fact_1409_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_1410_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ).

% max.strict_coboundedI1
tff(fact_1411_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ).

% max.strict_coboundedI2
tff(fact_1412_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q6: int,R4: int,Q4: int,R2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R2),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R4))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q6)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1413_unique__quotient__lemma,axiom,
    ! [B2: int,Q6: int,R4: int,Q4: int,R2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R4))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R2),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q6),Q4)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1414_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q4: int,R2: int,B6: int,Q6: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4)),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R2),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R4))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B6),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q6),Q4)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1415_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q4: int,R2: int,B6: int,Q6: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B6))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B6),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q6)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1416_q__pos__lemma,axiom,
    ! [B6: int,Q6: int,R4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q6)),R4)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B6))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q6)) ) ) ) ).

% q_pos_lemma
tff(fact_1417_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R2),B2))
         => ( divide_divide(int,A2,B2) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1418_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R2),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R2))
         => ( divide_divide(int,A2,B2) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1419_split__zdiv,axiom,
    ! [P: fun(int,bool),N: int,K: int] :
      ( pp(aa(int,bool,P,divide_divide(int,N,K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,zero_zero(int))) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I4: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => pp(aa(int,bool,P,I4)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I4: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => pp(aa(int,bool,P,I4)) ) ) ) ) ).

% split_zdiv
tff(fact_1420_zdiv__mono1,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),A6))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B2)),divide_divide(int,A6,B2))) ) ) ).

% zdiv_mono1
tff(fact_1421_zdiv__mono2,axiom,
    ! [A2: int,B6: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B6),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B2)),divide_divide(int,A2,B6))) ) ) ) ).

% zdiv_mono2
tff(fact_1422_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( divide_divide(int,I,K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1423_zdiv__mono1__neg,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),A6))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A6,B2)),divide_divide(int,A2,B2))) ) ) ).

% zdiv_mono1_neg
tff(fact_1424_zdiv__mono2__neg,axiom,
    ! [A2: int,B6: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B6))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B6),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B6)),divide_divide(int,A2,B2))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1425_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,K,L)))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ) ).

% div_int_pos_iff
tff(fact_1426_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),divide_divide(int,K,L))) ) ) ).

% div_positive_int
tff(fact_1427_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B2)),zero_zero(int))) ) ) ).

% div_nonneg_neg_le0
tff(fact_1428_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B2)),zero_zero(int))) ) ) ).

% div_nonpos_pos_le0
tff(fact_1429_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),divide_divide(int,I,K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1430_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,A2,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int))) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1431_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,A2,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1432_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),divide_divide(int,A2,B2)))
      <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1433_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A2,B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int))) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1434_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A2,B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A2)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1435_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,X,K)),X)) ) ) ).

% int_div_less_self
tff(fact_1436_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A2,B2)),zero_zero(int))) ) ) ).

% div_neg_pos_less0
tff(fact_1437_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A2,B2))) ) ) ) ).

% div_positive
tff(fact_1438_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1439_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1440_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),B2)) ) ) ).

% discrete
tff(fact_1441_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,R2: A,Q4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q4),R2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q4)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(num,A,numeral_numeral(A),L))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q4),R2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q4)),R2) ) ) ) ) ).

% divmod_step_eq
tff(fact_1442_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(1)
tff(fact_1443_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(2)
tff(fact_1444_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% unset_bit_0
tff(fact_1445_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),X))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_1446_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),X))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))),none(nat)) ) ) ) ).

% succ_greatereq_min
tff(fact_1447_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X),aa(nat,option(nat),some(nat),Mi),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))),none(nat)) ) ) ) ).

% pred_lesseq_max
tff(fact_1448_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X),aa(nat,option(nat),some(nat),Mi),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))) ) ) ) ) ).

% pred_less_length_list
tff(fact_1449_set__vebt_H__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_VEBT_set_vebt(T2) = aa(fun(nat,bool),set(nat),collect(nat),vEBT_vebt_member(T2)) ).

% set_vebt'_def
tff(fact_1450_Diff__idemp,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ).

% Diff_idemp
tff(fact_1451_Diff__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% Diff_iff
tff(fact_1452_DiffI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ) ) ).

% DiffI
tff(fact_1453_i0__less,axiom,
    ! [N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),N))
    <=> ( N != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_1454_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
        | pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),Q))) )
     => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_1455_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ab(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
        & pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),Q))) ) ) ).

% finite_Collect_disjI
tff(fact_1456_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_ac(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int1
tff(fact_1457_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_ad(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int4
tff(fact_1458_pred__empty,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_pred(T2,X) = none(nat) )
      <=> ( aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ae(vEBT_VEBT,fun(nat,fun(nat,bool)),T2),X)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_1459_succ__empty,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_succ(T2,X) = none(nat) )
      <=> ( aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_af(vEBT_VEBT,fun(nat,fun(nat,bool)),T2),X)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_1460_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2638667681897837118et_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% unset_bit_negative_int_iff
tff(fact_1461_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(set(set(A)),bool,finite_finite2(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ag(set(A),fun(set(A),bool),A3)))) ) ).

% finite_Collect_subsets
tff(fact_1462_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_ah(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int2
tff(fact_1463_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_ai(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int3
tff(fact_1464_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),K)))) ).

% finite_Collect_less_nat
tff(fact_1465_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ak(nat,fun(nat,bool)),K)))) ).

% finite_Collect_le_nat
tff(fact_1466_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = H )
           => ( ( vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ( Newlist = aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),Newnode) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                   => ( ( pp(vEBT_VEBT_minNull(Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                      & ( ~ pp(vEBT_VEBT_minNull(Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_1467_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = H )
           => ( ( vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( pp(vEBT_VEBT_minNull(Newnode))
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),Newnode) )
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(Sn)),none(nat)),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Sn)))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Sn))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_1468_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = H )
           => ( ( vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = L )
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
               => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),H)))),Ma))),Deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),Summary)) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_1469_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = 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) ) ) ) ).

% del_in_range
tff(fact_1470_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = L )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                 => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),H)))),Ma))),Deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),Summary)) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_1471_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = L )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),Newnode) )
                     => ( ( pp(vEBT_VEBT_minNull(Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                        & ( ~ pp(vEBT_VEBT_minNull(Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_1472_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT),Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = L )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),H),Newnode) )
                     => ( pp(vEBT_VEBT_minNull(Newnode))
                       => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(Sn)),none(nat)),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Sn)))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Sn))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_1473_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),X) = if(vEBT_VEBT,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),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),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))))))),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))))))),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),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),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))))))),Ma))),Deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),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),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_veb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m,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),Ma))),Deg,aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),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),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)) ) ) ) ) ).

% del_x_mia
tff(fact_1474_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),N)))
    <=> ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),M))
        & pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_1475_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),N),zero_zero(extended_enat))) ).

% not_iless0
tff(fact_1476_enat__less__induct,axiom,
    ! [P: fun(extended_enat,bool),N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),M2),N2))
             => pp(aa(extended_enat,bool,P,M2)) )
         => pp(aa(extended_enat,bool,P,N2)) )
     => pp(aa(extended_enat,bool,P,N)) ) ).

% enat_less_induct
tff(fact_1477_less__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ) ).

% less_set_def
tff(fact_1478_minus__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ).

% minus_set_def
tff(fact_1479_set__diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_al(set(A),fun(set(A),fun(A,bool)),A3),B3)) ).

% set_diff_eq
tff(fact_1480_DiffD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ).

% DiffD2
tff(fact_1481_DiffD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) ) ).

% DiffD1
tff(fact_1482_DiffE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% DiffE
tff(fact_1483_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X3: A,Xa3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),R))
        <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa3)),S2)) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_1484_Collect__restrict,axiom,
    ! [A: $tType,X6: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_am(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))),X6)) ).

% Collect_restrict
tff(fact_1485_prop__restrict,axiom,
    ! [A: $tType,X: A,Z5: set(A),X6: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),Z5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z5),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_am(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))))
       => pp(aa(A,bool,P,X)) ) ) ).

% prop_restrict
tff(fact_1486_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S2: set(A)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),R)),aTP_Lamp_a(set(A),fun(A,bool),S2)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R),S2)) ) ).

% pred_subset_eq
tff(fact_1487_less__eq__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ) ).

% less_eq_set_def
tff(fact_1488_Collect__subset,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_am(set(A),fun(fun(A,bool),fun(A,bool)),A3),P))),A3)) ).

% Collect_subset
tff(fact_1489_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),R: fun(A,fun(B,bool))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B3))
       => ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
             => ? [Xa: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa),B3))
                  & pp(aa(B,bool,aa(A,fun(B,bool),R,X4),Xa)) ) )
         => ? [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),B3))
              & ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_an(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),A3),R),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1490_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => ? [X_1: A] : pp(aa(A,bool,P,X_1)) ) ).

% not_finite_existsD
tff(fact_1491_empty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ao(A,bool)) ).

% empty_def
tff(fact_1492_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ap(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_1493_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Xa) = Xa ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Xa) = X2 ) ) ) ) ).

% max_def_raw
tff(fact_1494_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),S2)))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R),S2)) ) ).

% pred_subset_eq2
tff(fact_1495_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),bot_bot(set(product_prod(A,B))))) ) ).

% bot_empty_eq2
tff(fact_1496_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,bool),I: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ar(fun(nat,bool),fun(nat,fun(nat,bool)),P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_1497_finite__less__ub,axiom,
    ! [F2: fun(nat,nat),U: nat] :
      ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(nat,nat,F2,N2)))
     => pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_as(fun(nat,nat),fun(nat,fun(nat,bool)),F2),U)))) ) ).

% finite_less_ub
tff(fact_1498_set__vebt__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_set_vebt(T2) = aa(fun(nat,bool),set(nat),collect(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2)) ).

% set_vebt_def
tff(fact_1499_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_at(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_length_eq
tff(fact_1500_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_au(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_length_le
tff(fact_1501_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X4: int,K2: int] :
            ( pp(aa(int,bool,P,X4))
          <=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
        <=> ? [X3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X3),set_or1337092689740270186AtMost(int,one_one(int),D2)))
              & pp(aa(int,bool,P,X3)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1502_aset_I7_J,axiom,
    ! [D6: int,A3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6))) ) ) ) ).

% aset(7)
tff(fact_1503_aset_I5_J,axiom,
    ! [D6: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),A3))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),T2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6)),T2)) ) ) ) ) ).

% aset(5)
tff(fact_1504_aset_I4_J,axiom,
    ! [D6: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),A3))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X2 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6) != T2 ) ) ) ) ) ).

% aset(4)
tff(fact_1505_aset_I3_J,axiom,
    ! [D6: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A3))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X2 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6) = T2 ) ) ) ) ) ).

% aset(3)
tff(fact_1506_bset_I7_J,axiom,
    ! [D6: int,T2: int,B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),B3))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6))) ) ) ) ) ).

% bset(7)
tff(fact_1507_bset_I5_J,axiom,
    ! [D6: int,B3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B3))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),T2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6)),T2)) ) ) ) ).

% bset(5)
tff(fact_1508_bset_I4_J,axiom,
    ! [D6: int,T2: int,B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),B3))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X2 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6) != T2 ) ) ) ) ) ).

% bset(4)
tff(fact_1509_bset_I3_J,axiom,
    ! [D6: int,T2: int,B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int))),B3))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X2 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6) = T2 ) ) ) ) ) ).

% bset(3)
tff(fact_1510_aset_I8_J,axiom,
    ! [D6: int,A3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6))) ) ) ) ).

% aset(8)
tff(fact_1511_aset_I6_J,axiom,
    ! [D6: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A3))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),T2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6)),T2)) ) ) ) ) ).

% aset(6)
tff(fact_1512_bset_I8_J,axiom,
    ! [D6: int,T2: int,B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int))),B3))
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6))) ) ) ) ) ).

% bset(8)
tff(fact_1513_bset_I6_J,axiom,
    ! [D6: int,B3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),set_or1337092689740270186AtMost(int,one_one(int),D6)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B3))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),T2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D6)),T2)) ) ) ) ).

% bset(6)
tff(fact_1514_cpmi,axiom,
    ! [D6: int,P: fun(int,bool),P3: fun(int,bool),B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X4),Z4))
           => ( pp(aa(int,bool,P,X4))
            <=> pp(aa(int,bool,P3,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                 => ! [Xb: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),B3))
                     => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
             => ( pp(aa(int,bool,P,X4))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D6))) ) )
         => ( ! [X4: int,K2: int] :
                ( pp(aa(int,bool,P3,X4))
              <=> pp(aa(int,bool,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D6)))) )
           => ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
            <=> ( ? [X3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X3),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                    & pp(aa(int,bool,P3,X3)) )
                | ? [X3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X3),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                    & ? [Xa3: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),B3))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X3))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_1515_cppi,axiom,
    ! [D6: int,P: fun(int,bool),P3: fun(int,bool),A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D6))
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),X4))
           => ( pp(aa(int,bool,P,X4))
            <=> pp(aa(int,bool,P3,X4)) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                 => ! [Xb: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb),A3))
                     => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
             => ( pp(aa(int,bool,P,X4))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D6))) ) )
         => ( ! [X4: int,K2: int] :
                ( pp(aa(int,bool,P3,X4))
              <=> pp(aa(int,bool,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D6)))) )
           => ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
            <=> ( ? [X3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X3),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                    & pp(aa(int,bool,P3,X3)) )
                | ? [X3: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X3),set_or1337092689740270186AtMost(int,one_one(int),D6)))
                    & ? [Xa3: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),A3))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa3),X3))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_1516_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V2),TreeList,S),X)
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_1517_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd),X)
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_1518_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
             => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
                & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
                 => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                     => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_1519_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V2),TreeList,Vc),X)
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_1520_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa2)
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( ( ( Xa2 = zero_zero(nat) )
               => pp(A4) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( ( ( Xa2 = one_one(nat) )
                   => pp(B4) )
                  & ( Xa2 = one_one(nat) ) ) ) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S4: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4)
               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_1521_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_V5719532721284313246member(X,Xa2)
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( ( Xa2 = zero_zero(nat) )
                 => pp(A4) )
                & ( ( Xa2 != zero_zero(nat) )
                 => ( ( ( Xa2 = one_one(nat) )
                     => pp(B4) )
                    & ( Xa2 = one_one(nat) ) ) ) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S4: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4)
             => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_1522_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa2)
      <=> pp(Y) )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B4) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S4: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4)
               => ( pp(Y)
                <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_1523_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_membermima(X,Xa2)
     => ( ! [Mi3: nat,Ma3: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xa2 = Mi3 )
                | ( Xa2 = Ma3 ) ) )
       => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2)
             => ~ ( ( Xa2 = Mi3 )
                  | ( Xa2 = Ma3 )
                  | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) )
         => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)
               => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_1524_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( ( Xa2 = zero_zero(nat) )
                 => pp(A4) )
                & ( ( Xa2 != zero_zero(nat) )
                 => ( ( ( Xa2 = one_one(nat) )
                     => pp(B4) )
                    & ( Xa2 = one_one(nat) ) ) ) ) )
       => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary3: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)
             => ~ ( ( Xa2 != Mi3 )
                 => ( ( Xa2 != Ma3 )
                   => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                           => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                               => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) )
                              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_1525_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa2)
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2)
               => ( ( Xa2 = Mi3 )
                  | ( Xa2 = Ma3 ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2)
                 => ( ( Xa2 = Mi3 )
                    | ( Xa2 = Ma3 )
                    | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) )
             => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)
                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_1526_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa2)
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi3 )
                      | ( Xa2 = Ma3 ) ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa2 = Mi3 )
                        | ( Xa2 = Ma3 )
                        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) )
             => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_1527_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(vEBT_VEBT,fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(bool,bool,fNot,fdisj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma)))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),X,Mi)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X)),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary)) ).

% vebt_insert.simps(5)
tff(fact_1528_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( ( ( Xa2 = zero_zero(nat) )
               => pp(A4) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( ( ( Xa2 = one_one(nat) )
                   => pp(B4) )
                  & ( Xa2 = one_one(nat) ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
           => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary3: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)
                   => ( ( Xa2 != Mi3 )
                     => ( ( Xa2 != Ma3 )
                       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                   => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) )
                                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_1529_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B4) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => pp(Y) )
         => ( ( ? [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
             => pp(Y) )
           => ( ( ? [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => pp(Y) )
             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary3: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)
                   => ( pp(Y)
                    <=> ~ ( ( Xa2 != Mi3 )
                         => ( ( Xa2 != Ma3 )
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                               => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                                  & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                       => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) )
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_1530_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ~ ( ( ( Xa2 = zero_zero(nat) )
                 => ( Y = vEBT_Leaf(fTrue,B4) ) )
                & ( ( Xa2 != zero_zero(nat) )
                 => ( ( ( Xa2 = one_one(nat) )
                     => ( Y = vEBT_Leaf(A4,fTrue) ) )
                    & ( ( Xa2 != one_one(nat) )
                     => ( Y = vEBT_Leaf(A4,B4) ) ) ) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info2,zero_zero(nat),Ts2,S4) )
             => ( Y != vEBT_Node(Info2,zero_zero(nat),Ts2,S4) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4) )
               => ( Y != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4) ) )
           => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3) )
                 => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),Xa2)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3) ) )
             => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                   => ( Y != if(vEBT_VEBT,fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(bool,bool,fNot,fdisj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Mi3),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Ma3)))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Xa2,Mi3)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2)),Ma3))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summary3,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summary3)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_1531_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,option(nat),some(nat),Mi) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))),none(nat)) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_1532_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = aa(nat,option(nat),some(nat),Ma) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X),aa(nat,option(nat),some(nat),Mi),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))),none(nat)) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_1533_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X)) )
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X)) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary),X) = 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) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_1534_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( ! [A4: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A4,B4) )
           => ( ( Xa2 = zero_zero(nat) )
             => ( Y != vEBT_Leaf(fFalse,B4) ) ) )
       => ( ! [A4: bool] :
              ( ? [B4: bool] : X = vEBT_Leaf(A4,B4)
             => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf(A4,fFalse) ) ) )
         => ( ! [A4: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A4,B4) )
               => ( ? [N2: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N2))
                 => ( Y != vEBT_Leaf(A4,B4) ) ) )
           => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3) ) )
             => ( ! [Mi3: nat,Ma3: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),TrLst2,Smry2) )
                   => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),TrLst2,Smry2) ) )
               => ( ! [Mi3: nat,Ma3: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                     => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) ) )
                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                                | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2)) )
                             => ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) ) )
                            & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                                  | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2)) )
                             => ( ( ( ( Xa2 = Mi3 )
                                    & ( Xa2 = Ma3 ) )
                                 => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) ) )
                                & ( ~ ( ( Xa2 = Mi3 )
                                      & ( Xa2 = Ma3 ) )
                                 => ( Y = 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) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_1535_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( ! [Uu2: bool,B4: bool] :
            ( ( X = vEBT_Leaf(Uu2,B4) )
           => ( ( Xa2 = zero_zero(nat) )
             => ~ ( ( pp(B4)
                   => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                  & ( ~ pp(B4)
                   => ( Y = none(nat) ) ) ) ) )
       => ( ( ? [Uv2: bool,Uw2: bool] : X = vEBT_Leaf(Uv2,Uw2)
           => ( ? [N2: nat] : Xa2 = aa(nat,nat,suc,N2)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)
               => ( Y != none(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : X = 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)),Vg2,Vh2)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                     => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                           => ( Y = aa(nat,option(nat),some(nat),Mi3) ) )
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                           => ( Y = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_succ(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))),none(nat)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_1536_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( ( Xa2 = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A4: bool] :
              ( ? [Uw2: bool] : X = vEBT_Leaf(A4,Uw2)
             => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ ( ( pp(A4)
                     => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                    & ( ~ pp(A4)
                     => ( Y = none(nat) ) ) ) ) )
         => ( ! [A4: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A4,B4) )
               => ( ? [Va2: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
                 => ~ ( ( pp(B4)
                       => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                      & ( ~ pp(B4)
                       => ( ( pp(A4)
                           => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                          & ( ~ pp(A4)
                           => ( Y = none(nat) ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != none(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)
                 => ( Y != none(nat) ) )
               => ( ( ? [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : X = 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)),Vh2,Vi2)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                       => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                             => ( Y = aa(nat,option(nat),some(nat),Ma3) ) )
                            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                             => ( Y = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),Xa2),aa(nat,option(nat),some(nat),Mi3),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_pred(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))),none(nat)) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_1537_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(set(complex),bool,finite_finite2(complex),aa(fun(complex,bool),set(complex),collect(complex),aa(complex,fun(complex,bool),aTP_Lamp_av(nat,fun(complex,fun(complex,bool)),N),C2)))) ) ).

% finite_nth_roots
tff(fact_1538_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aw(nat,fun(A,bool),N)))) ) ) ).

% finite_roots_unity
tff(fact_1539_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,B4: bool] :
              ( ( X = vEBT_Leaf(Uu2,B4) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( ( pp(B4)
                     => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                    & ( ~ pp(B4)
                     => ( Y = none(nat) ) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,B4)),zero_zero(nat))) ) ) )
         => ( ! [Uv2: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(Uv2,Uw2) )
               => ! [N2: nat] :
                    ( ( Xa2 = aa(nat,nat,suc,N2) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uv2,Uw2)),aa(nat,nat,suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xa2)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)),Xa2)) ) )
               => ( ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( X = 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)),Vg2,Vh2) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xa2)) ) )
                 => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                       => ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                             => ( Y = aa(nat,option(nat),some(nat),Mi3) ) )
                            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                             => ( Y = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_succ(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))),none(nat)) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),Xa2)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_1540_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),zero_zero(nat))) ) ) )
         => ( ! [A4: bool,Uw2: bool] :
                ( ( X = vEBT_Leaf(A4,Uw2) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( ( pp(A4)
                       => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                      & ( ~ pp(A4)
                       => ( Y = none(nat) ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,Uw2)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A4,B4) )
                 => ! [Va2: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( ( pp(B4)
                           => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                          & ( ~ pp(B4)
                           => ( ( pp(A4)
                               => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                              & ( ~ pp(A4)
                               => ( Y = none(nat) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xa2)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)),Xa2)) ) )
                 => ( ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( X = 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)),Vh2,Vi2) )
                       => ( ( Y = none(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xa2)) ) )
                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                         => ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                               => ( Y = aa(nat,option(nat),some(nat),Ma3) ) )
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                               => ( Y = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),Xa2),aa(nat,option(nat),some(nat),Mi3),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_pred(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary3,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))))),none(nat)) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_1541_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf(fFalse,B4) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),zero_zero(nat))) ) ) )
         => ( ! [A4: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A4,B4) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf(A4,fFalse) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A4,B4) )
                 => ! [N2: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                     => ( ( Y = vEBT_Leaf(A4,B4) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),aa(nat,nat,suc,aa(nat,nat,suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3)),Xa2)) ) )
               => ( ! [Mi3: nat,Ma3: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),TrLst2,Smry2) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),TrLst2,Smry2)),Xa2)) ) )
                 => ( ! [Mi3: nat,Ma3: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2)),Xa2)) ) )
                   => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                         => ( ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                                  | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2)) )
                               => ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) ) )
                              & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                                    | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2)) )
                               => ( ( ( ( Xa2 = Mi3 )
                                      & ( Xa2 = Ma3 ) )
                                   => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) ) )
                                  & ( ~ ( ( Xa2 = Mi3 )
                                        & ( Xa2 = Ma3 ) )
                                   => ( Y = 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,aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary3)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Mi3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary3))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary3)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),Summary3)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_1542_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( ( ( Xa2 = zero_zero(nat) )
                   => ( Y = vEBT_Leaf(fTrue,B4) ) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => ( Y = vEBT_Leaf(A4,fTrue) ) )
                      & ( ( Xa2 != one_one(nat) )
                       => ( Y = vEBT_Leaf(A4,B4) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),Xa2)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info2,zero_zero(nat),Ts2,S4) )
               => ( ( Y = vEBT_Node(Info2,zero_zero(nat),Ts2,S4) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S4)),Xa2)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4) )
                 => ( ( Y = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4)),Xa2)) ) )
             => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3) )
                   => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),Xa2)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3)),Xa2)) ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                     => ( ( Y = if(vEBT_VEBT,fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(bool,bool,fNot,fdisj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Mi3),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Ma3)))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Xa2,Mi3)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2)),Ma3))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),aa(list(vEBT_VEBT),fun(nat,fun(vEBT_VEBT,list(vEBT_VEBT))),list_update(vEBT_VEBT),TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summary3,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3),Mi3,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summary3)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_1543_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,N: nat] :
      ( ! [N2: nat] : aa(nat,A,F2,aa(nat,nat,suc,N2)) = aa(nat,A,H,N2)
     => ( ( aa(nat,A,F2,zero_zero(nat)) = G )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,F2,N) = G ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,F2,N) = aa(nat,A,H,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ) ) ).

% Suc_if_eq
tff(fact_1544_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or1337092689740270186AtMost(int,L,U))) ).

% finite_atLeastAtMost_int
tff(fact_1545_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_1546_bot2E,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : ~ pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X),Y)) ).

% bot2E
tff(fact_1547_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B4) )
                        & ( Xa2 = one_one(nat) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xa2)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xa2)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ pp(Y)
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa2)) ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                     => ( ( pp(Y)
                        <=> ( ( Xa2 != Mi3 )
                           => ( ( Xa2 != Ma3 )
                             => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                                & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                                 => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                                    & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                                     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                         => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) )
                                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_1548_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),Xa2))
               => ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B4) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xa2)) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xa2)) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa2)) )
               => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),Xa2))
                       => ( ( Xa2 != Mi3 )
                         => ( ( Xa2 != Ma3 )
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                               => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                                  & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                       => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) )
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_1549_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),Xa2))
               => ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A4) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B4) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xa2)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4)),Xa2))
                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_1550_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_V5719532721284313246member(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),Xa2))
               => ~ ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B4) )
                        & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4)),Xa2))
                 => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_1551_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa2)
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B4) )
                        & ( Xa2 = one_one(nat) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),Xa2)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xa2)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S4)),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1552_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A4,B4)),Xa2))
               => ~ ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A4) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B4) )
                        & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ~ ! [Mi3: nat,Ma3: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary3)),Xa2))
                 => ~ ( ( Xa2 != Mi3 )
                     => ( ( Xa2 != Ma3 )
                       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi3))
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),Xa2))
                               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                                   => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) )
                                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_1553_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa2)
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xa2)) ) )
           => ( ! [Mi3: nat,Ma3: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa2 = Mi3 )
                        | ( Xa2 = Ma3 ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2)),Xa2)) ) )
             => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa2 = Mi3 )
                          | ( Xa2 = Ma3 )
                          | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2)),Xa2)) ) )
               => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_1554_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xa2)) )
           => ( ! [Mi3: nat,Ma3: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2)),Xa2))
                   => ( ( Xa2 = Mi3 )
                      | ( Xa2 = Ma3 ) ) ) )
             => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2) )
                   => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2)),Xa2))
                     => ( ( Xa2 = Mi3 )
                        | ( Xa2 = Ma3 )
                        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) )
               => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),Xa2))
                       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_1555_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_membermima(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Mi3: nat,Ma3: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),zero_zero(nat),Va3,Vb2)),Xa2))
               => ~ ( ( Xa2 = Mi3 )
                    | ( Xa2 = Ma3 ) ) ) )
         => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc2)),Xa2))
                 => ~ ( ( Xa2 = Mi3 )
                      | ( Xa2 = Ma3 )
                      | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) )
           => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),Xa2))
                   => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_1556_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1557_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1558_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1559_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q4: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q4),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_1560_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arsinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_1561_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_1562_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q4: int,R2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),zero_zero(int)))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_1563_low__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_low(X,N) = modulo_modulo(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% low_def
tff(fact_1564_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_1565_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_1566_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_1567_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_1568_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( modulo_modulo(nat,M,N) = M ) ) ).

% mod_less
tff(fact_1569_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),B2) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_1570_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),B2) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_1571_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_1572_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : modulo_modulo(A,A2,one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_1573_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : divide_divide(A,modulo_modulo(A,A2,B2),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_1574_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : divide_divide(A,modulo_modulo(A,A2,B2),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_1575_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_1576_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1577_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1578_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1579_add__self__mod__2,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_1580_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_1581_unique__quotient,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int,Q6: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R4))
       => ( Q4 = Q6 ) ) ) ).

% unique_quotient
tff(fact_1582_unique__remainder,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int,Q6: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_1583_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N)),M)) ).

% mod_less_eq_dividend
tff(fact_1584_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_1585_div__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2))
     => ( divide_divide(int,K,L) = Q4 ) ) ).

% div_int_unique
tff(fact_1586_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),modulo_modulo(A,A2,B2)),A2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1587_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A2,B2)),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1588_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = A2 )
        <=> ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1589_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) = N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = zero_zero(nat) ) )
      & ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) != N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = aa(nat,nat,suc,modulo_modulo(nat,M,N)) ) ) ) ).

% mod_Suc
tff(fact_1590_mod__induct,axiom,
    ! [P: fun(nat,bool),N: nat,P2: nat,M: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),P2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),P2))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),P2))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,modulo_modulo(nat,aa(nat,nat,suc,N2),P2))) ) )
           => pp(aa(nat,bool,P,M)) ) ) ) ) ).

% mod_induct
tff(fact_1591_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),modulo_modulo(nat,M,N)),N)) ) ).

% mod_less_divisor
tff(fact_1592_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [M5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M5),zero_zero(nat)))
     => ( ! [M5: nat,N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N2),modulo_modulo(nat,M5,N2)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M5),N2)) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_1593_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,aa(nat,nat,suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_1594_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_1595_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N) ) ) ).

% mod_geq
tff(fact_1596_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( modulo_modulo(nat,M,N) = M ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N) ) ) ) ).

% mod_if
tff(fact_1597_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N) ) ) ).

% le_mod_geq
tff(fact_1598_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A2,B2))) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1599_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( modulo_modulo(A,A2,B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1600_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q4: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q4))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1601_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_1602_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N)),N)) ) ).

% mod_le_divisor
tff(fact_1603_div__less__mono,axiom,
    ! [A3: nat,B3: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( modulo_modulo(nat,A3,N) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B3,N) = zero_zero(nat) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,A3,N)),divide_divide(nat,B3,N))) ) ) ) ) ).

% div_less_mono
tff(fact_1604_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q4) = modulo_modulo(nat,N,Q4) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ~ ! [S4: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),S4)) ) ) ).

% mod_eq_nat1E
tff(fact_1605_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q4) = modulo_modulo(nat,N,Q4) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ~ ! [S4: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),S4)) ) ) ).

% mod_eq_nat2E
tff(fact_1606_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y,N) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
       => ? [Q3: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1607_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_1608_split__mod,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,modulo_modulo(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,M)) )
        & ( ( N != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I4)),J3) )
               => pp(aa(nat,bool,P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_1609_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
         => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,divide_divide(A,A2,B2),C2))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1610_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1611_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2))
           => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)) = modulo_modulo(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1612_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_1613_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,A3,N)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),modulo_modulo(nat,B3,N)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),divide_divide(nat,B3,N))) ) ) ).

% verit_le_mono_div
tff(fact_1614_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))) = divide_divide(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1615_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1616_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),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),Q4)),R2) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R2))
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R2),L)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),R2))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R2),zero_zero(int))) ) )
            & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( Q4 = zero_zero(int) ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1617_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = modulo_modulo(A,X,M) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_1618_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2) = modulo_modulo(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1619_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))),one_one(A)) = divide_divide(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1620_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mult_less_iff1
tff(fact_1621_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q4: int,R2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_1622_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),N) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),divide_divide(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_1623_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I6: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ax(set(B),fun(fun(B,A),fun(B,bool)),I6),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ax(set(B),fun(fun(B,A),fun(B,bool)),I6),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ay(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I6),X),Y)))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_1624_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I6: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_az(set(B),fun(fun(B,A),fun(B,bool)),I6),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_az(set(B),fun(fun(B,A),fun(B,bool)),I6),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ba(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I6),X),Y)))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_1625_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_1626_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,bool))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( ! [A4: real,B4: real,C4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),P,A4),B4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),P,B4),C4))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A4),B4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B4),C4))
                 => pp(aa(real,bool,aa(real,fun(real,bool),P,A4),C4)) ) ) ) )
       => ( ! [X4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2))
               => ? [D4: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D4))
                    & ! [A4: real,B4: real] :
                        ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A4),X4))
                          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B4))
                          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B4),A4)),D4)) )
                       => pp(aa(real,bool,aa(real,fun(real,bool),P,A4),B4)) ) ) ) )
         => pp(aa(real,bool,aa(real,fun(real,bool),P,A2),B2)) ) ) ) ).

% Bolzano
tff(fact_1627_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),one_one(nat)))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_1628_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),zero_zero(A))) ) ).

% dvd_0_right
tff(fact_1629_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A2))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_1630_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( M = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_1631_dvd__1__left,axiom,
    ! [K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K)) ).

% dvd_1_left
tff(fact_1632_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( ( K = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1633_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( C2 = zero_zero(A) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1634_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( C2 = zero_zero(A) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1635_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1636_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1637_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
         => ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1638_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1639_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1640_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A2: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ) ).

% pow_divides_pow_iff
tff(fact_1641_length__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_product
tff(fact_1642_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W2))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1643_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1644_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% power_less_zero_eq
tff(fact_1645_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_1646_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ).

% even_diff_nat
tff(fact_1647_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W2))))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W2) = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
              & ( A2 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1648_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% even_power
tff(fact_1649_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W2)))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W2)))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1650_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A))))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1651_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = divide_divide(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_1652_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_1653_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A2))
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1654_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),zero_zero(nat))) ).

% gcd_nat.extremum
tff(fact_1655_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A2))
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_1656_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A2))
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_1657_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A2),zero_zero(nat)))
        & ( A2 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_1658_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A2))
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_1659_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).

% dvd_diff_nat
tff(fact_1660_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        | ( ( L = zero_zero(int) )
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) )
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L)) ) ) ).

% mod_int_pos_iff
tff(fact_1661_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bb(A,fun(A,bool),A2))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bb(A,fun(A,bool),B2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2)) ) ) ).

% subset_divisors_dvd
tff(fact_1662_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bb(A,fun(A,bool),A2))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bb(A,fun(A,bool),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2)) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1663_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),one_one(A))) ) ).

% not_is_unit_0
tff(fact_1664_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z3: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X2))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% pinf(9)
tff(fact_1665_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z3: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X2))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% pinf(10)
tff(fact_1666_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z3: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Z3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% minf(9)
tff(fact_1667_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z3: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Z3))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% minf(10)
tff(fact_1668_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1669_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2)) ) ) ).

% mod_0_imp_dvd
tff(fact_1670_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
        <=> ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1671_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2)) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1672_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,N: nat,M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M))) ) ) ) ).

% dvd_power_le
tff(fact_1673_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat,B2: A,M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),B2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),B2)) ) ) ) ).

% power_le_dvd
tff(fact_1674_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% le_imp_power_dvd
tff(fact_1675_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,K,L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1676_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),modulo_modulo(int,K,L))) ) ).

% neg_mod_bound
tff(fact_1677_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_1678_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) ) ) ).

% dvd_pos_nat
tff(fact_1679_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% dvd_minus_self
tff(fact_1680_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_1681_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_1682_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M)) ) ) ) ).

% dvd_diffD
tff(fact_1683_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N))
       => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M)) ) ) ).

% zdvd_not_zless
tff(fact_1684_mod__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2))
     => ( modulo_modulo(int,K,L) = R2 ) ) ).

% mod_int_unique
tff(fact_1685_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aTP_Lamp_bc(int,fun(int,bool),I)))) ) ).

% finite_divisors_int
tff(fact_1686_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,bool),L: A] :
          ( ? [X3: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X3)))
        <=> ? [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),zero_zero(A))))
              & pp(aa(A,bool,P,X3)) ) ) ) ).

% unity_coeff_ex
tff(fact_1687_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C4) ) ) ) ).

% unit_dvdE
tff(fact_1688_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),D2))
               => ( ( divide_divide(A,B2,A2) = divide_divide(A,D2,C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1689_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),divide_divide(A,B2,C2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1690_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),divide_divide(A,A2,B2)),C2))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1691_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B2))
           => ( ( divide_divide(A,B2,A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1692_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1693_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1694_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1695_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,K,L)),zero_zero(int))) ) ).

% neg_mod_sign
tff(fact_1696_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1697_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( modulo_modulo(int,I,K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I)) ) ) ) ).

% zmod_trivial_iff
tff(fact_1698_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A2,B2)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,A2,B2)),B2)) ) ) ).

% pos_mod_conj
tff(fact_1699_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,A2,B2)),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),modulo_modulo(int,A2,B2))) ) ) ).

% neg_mod_conj
tff(fact_1700_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).

% dvd_imp_le
tff(fact_1701_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1702_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_1703_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => ( ( modulo_modulo(int,A3,N) = zero_zero(int) )
         => ( ( modulo_modulo(int,B3,N) = zero_zero(int) )
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A3,N)),divide_divide(int,B3,N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1704_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D3: nat,X4: nat,Y2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A2))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),B2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1705_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z),N))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_1706_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,N)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1707_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( ( modulo_modulo(nat,M,Q4) = modulo_modulo(nat,N,Q4) )
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1708_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,K,L)),modulo_modulo(int,K,L))) ).

% eucl_rel_int
tff(fact_1709_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod(nat,nat)] :
      ~ ! [K2: nat,M5: nat] : X != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K2),M5) ).

% prod_decode_aux.cases
tff(fact_1710_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_bd(nat,fun(nat,bool),M)))) ) ).

% finite_divisors_nat
tff(fact_1711_full__exhaustive__int_H_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int))] :
      ~ ! [F3: fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D3: int,I2: int] : X != aa(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int)),aa(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),fun(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int))),product_Pair(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D3),I2)) ).

% full_exhaustive_int'.cases
tff(fact_1712_exhaustive__int_H_Ocases,axiom,
    ! [X: product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int))] :
      ~ ! [F3: fun(int,option(product_prod(bool,list(code_term)))),D3: int,I2: int] : X != aa(product_prod(int,int),product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int)),aa(fun(int,option(product_prod(bool,list(code_term)))),fun(product_prod(int,int),product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int))),product_Pair(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D3),I2)) ).

% exhaustive_int'.cases
tff(fact_1713_small__lazy_H_Ocases,axiom,
    ! [X: product_prod(int,int)] :
      ~ ! [D3: int,I2: int] : X != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D3),I2) ).

% small_lazy'.cases
tff(fact_1714_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),zero_zero(A))) ) ).

% even_zero
tff(fact_1715_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_1716_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_1717_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A)))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B4: A] :
                  ( ( B4 != zero_zero(A) )
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B4),one_one(A)))
                   => ( ( divide_divide(A,one_one(A),A2) = B4 )
                     => ( ( divide_divide(A,one_one(A),B4) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B4) = one_one(A) )
                         => ( divide_divide(A,C2,A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1718_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A)))
              | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ) ).

% dvd_power_iff
tff(fact_1719_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            | ( X = one_one(A) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))) ) ) ).

% dvd_power
tff(fact_1720_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1721_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1722_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_1723_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_1724_dvd__minus__add,axiom,
    ! [Q4: nat,N: nat,R2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q4),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Q4)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M)),Q4)))) ) ) ) ).

% dvd_minus_add
tff(fact_1725_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_1726_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),R2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R2),M))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),R2)))
         => ( modulo_modulo(nat,M,N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_1727_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1728_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A,B2: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono_odd
tff(fact_1729_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% odd_pos
tff(fact_1730_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_1731_split__zmod,axiom,
    ! [P: fun(int,bool),N: int,K: int] :
      ( pp(aa(int,bool,P,modulo_modulo(int,N,K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,N)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I4: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => pp(aa(int,bool,P,J3)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I4: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => pp(aa(int,bool,P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_1732_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R2),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R2))
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1733_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R2) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R2),B2))
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1734_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2638667681897837118et_bit(A,M,A2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_1735_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se5668285175392031749et_bit(A,M,A2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_1736_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ) ).

% mod2_eq_if
tff(fact_1737_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) ) )
         => ~ ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
             => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_1738_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N))) ) ) ).

% zero_le_even_power
tff(fact_1739_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% zero_le_odd_power
tff(fact_1740_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ) ).

% zero_le_power_eq
tff(fact_1741_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A3,N)),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,B3,N)),zero_zero(int)),one_one(int),zero_zero(int)))),divide_divide(int,B3,N))) ) ) ).

% verit_le_mono_div_int
tff(fact_1742_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,divide_divide(int,N,K)),modulo_modulo(int,N,K)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I4),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_1743_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,divide_divide(int,N,K)),modulo_modulo(int,N,K)))
      <=> ! [I4: int,J3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I4),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_1744_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N2: nat] : X != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_1745_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)))
        <=> ( ( N = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
              & ( A2 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ) ).

% zero_less_power_eq
tff(fact_1746_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% even_mask_div_iff'
tff(fact_1747_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1748_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% even_mod_4_div_2
tff(fact_1749_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% even_mask_div_iff
tff(fact_1750_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_1751_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),replicate(vEBT_VEBT),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) )
                    & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),replicate(vEBT_VEBT),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_1752_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X22: A] : aa(option(A),nat,size_option(A,X),aa(A,option(A),some(A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_1753_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% diff_shunt_var
tff(fact_1754_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_be(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_1755_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).

% take_bit_rec
tff(fact_1756_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : size_num(aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_1757_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),replicate(vEBT_VEBT),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),replicate(vEBT_VEBT),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_1758_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se8732182000553998342ip_bit(A,M,A2)))
        <=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_1759_intind,axiom,
    ! [A: $tType,I: nat,N: nat,P: fun(A,bool),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
     => ( pp(aa(A,bool,P,X))
       => pp(aa(A,bool,P,aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),I))) ) ) ).

% intind
tff(fact_1760_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_1761_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se8732182000553998342ip_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% flip_bit_negative_int_iff
tff(fact_1762_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,X: A,N: nat,Y: A] :
      ( ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M),X) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero(nat) )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1763_length__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),nat,size_size(list(A)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = N ).

% length_replicate
tff(fact_1764_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),A2: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),F2,A2),B2) ).

% case_prod_conv
tff(fact_1765_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),A2) = zero_zero(A) ) ).

% take_bit_0
tff(fact_1766_in__set__replicate,axiom,
    ! [A: $tType,X: A,N: nat,Y: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_1767_Bex__set__replicate,axiom,
    ! [A: $tType,N: nat,A2: A,P: fun(A,bool)] :
      ( ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),A2))))
          & pp(aa(A,bool,P,X3)) )
    <=> ( pp(aa(A,bool,P,A2))
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_1768_Ball__set__replicate,axiom,
    ! [A: $tType,N: nat,A2: A,P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),A2))))
         => pp(aa(A,bool,P,X3)) )
    <=> ( pp(aa(A,bool,P,A2))
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_1769_nth__replicate,axiom,
    ! [A: $tType,I: nat,N: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
     => ( aa(nat,A,nth(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),I) = X ) ) ).

% nth_replicate
tff(fact_1770_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_1771_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)))
        <=> ( ( N = zero_zero(nat) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ) ).

% even_take_bit_eq
tff(fact_1772_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),A2) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_0
tff(fact_1773_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_1774_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M,A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M,A2)) = bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_1775_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_1776_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q4)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),Q4))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_1777_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_1778_old_Oprod_Ocase,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(A,fun(B,C)),X1: A,X22: B] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = aa(B,C,aa(A,fun(B,C),F2,X1),X22) ).

% old.prod.case
tff(fact_1779_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q4: product_prod(A,B),F2: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P2: product_prod(A,B)] :
      ( ! [X4: A,Y2: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2) = Q4 )
         => ( aa(B,C,aa(A,fun(B,C),F2,X4),Y2) = aa(B,C,aa(A,fun(B,C),G,X4),Y2) ) )
     => ( ( P2 = Q4 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),P2) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q4) ) ) ) ).

% split_cong
tff(fact_1780_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,bool),P: fun(B,fun(C,A)),Z: product_prod(B,C)] :
      ( pp(aa(A,bool,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z)))
     => ~ ! [X4: B,Y2: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y2) )
           => ~ pp(aa(A,bool,Q,aa(C,A,aa(B,fun(C,A),P,X4),Y2))) ) ) ).

% case_prodE2
tff(fact_1781_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_bf(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2)) = F2 ).

% case_prod_eta
tff(fact_1782_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X4: A,Y2: B] : aa(B,C,aa(A,fun(B,C),F2,X4),Y2) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2) = G ) ) ).

% cond_case_prod_eta
tff(fact_1783_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_1784_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),zero_zero(int))) ).

% not_take_bit_negative
tff(fact_1785_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% take_bit_int_greater_self_iff
tff(fact_1786_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se2638667681897837118et_bit(A,M,A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se2638667681897837118et_bit(A,M,A2)) = bit_se2638667681897837118et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_1787_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se5668285175392031749et_bit(A,M,A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se5668285175392031749et_bit(A,M,A2)) = bit_se5668285175392031749et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_1788_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
     => ( ! [Y2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),aa(list(A),set(A),set2(A),Xs)))
           => ( Y2 = X ) )
       => ( Xs = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ) ) ) ).

% replicate_eqI
tff(fact_1789_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( X4 = X ) )
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),Xs)),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1790_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_1791_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% take_bit_nat_less_exp
tff(fact_1792_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_1793_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% take_bit_int_less_exp
tff(fact_1794_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_1795_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),A2)) ) ) ).

% take_bit_eq_0_iff
tff(fact_1796_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_1797_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_1798_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_1799_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_1800_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_1801_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_1802_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_1803_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) ) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) ) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_1804_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : aa(option(A),nat,size_option(A,X),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_1805_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_bg(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_1806_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_bh(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_1807_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_bi(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_1808_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero(nat) )
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) ) )
      & ( ~ ( ( N = zero_zero(nat) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = aa(product_prod(nat,nat),product_prod(nat,nat),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_bj(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_1809_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),replicate(vEBT_VEBT),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) )
                      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(vEBT_VEBT,list(vEBT_VEBT),aa(nat,fun(vEBT_VEBT,list(vEBT_VEBT)),replicate(vEBT_VEBT),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_1810_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% flip_bit_0
tff(fact_1811_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),nat_set_decode(X)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X)) ) ).

% set_decode_0
tff(fact_1812_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(nat),nat,nat_set_encode,A3)))
      <=> ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) ) ) ).

% even_set_encode_iff
tff(fact_1813_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_1814_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,bool))] :
      ( ! [A4: A,B4: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => pp(aa(B,bool,aa(A,fun(B,bool),C2,A4),B4)) )
     => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C2),P2)) ) ).

% case_prodI2
tff(fact_1815_case__prodI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,bool)),A2: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),F2,A2),B2))
     => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2))) ) ).

% case_prodI
tff(fact_1816_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B4: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Z),aa(B,set(C),aa(A,fun(B,set(C)),C2,A4),B4))) )
     => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Z),aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P2))) ) ).

% mem_case_prodI2
tff(fact_1817_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(C,set(A),aa(B,fun(C,set(A)),C2,A2),B2)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)))) ) ).

% mem_case_prodI
tff(fact_1818_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(C,bool))),X: C] :
      ( ! [A4: A,B4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) = P2 )
         => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,A4),B4),X)) )
     => pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C2),P2),X)) ) ).

% case_prodI2'
tff(fact_1819_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( pp(P)
           => pp(Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_1820_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fFalse) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_1821_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = zero_zero(A) )
        <=> ~ pp(P) ) ) ).

% of_bool_eq_0_iff
tff(fact_1822_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( ~ pp(P)
            & pp(Q) ) ) ) ).

% of_bool_less_iff
tff(fact_1823_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P)))
        <=> pp(P) ) ) ).

% zero_less_of_bool_iff
tff(fact_1824_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A)))
        <=> ~ pp(P) ) ) ).

% of_bool_less_one_iff
tff(fact_1825_Suc__0__mod__eq,axiom,
    ! [N: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),aa(nat,nat,suc,zero_zero(nat))))) ).

% Suc_0_mod_eq
tff(fact_1826_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_1827_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_1828_set__encode__inverse,axiom,
    ! [A3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A3)) = A3 ) ) ).

% set_encode_inverse
tff(fact_1829_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_1830_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_1831_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_1832_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: bool] : divide_divide(A,aa(bool,A,zero_neq_one_of_bool(A),B2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_1833_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit0,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_1834_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% bits_1_div_exp
tff(fact_1835_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% one_div_2_pow_eq
tff(fact_1836_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% take_bit_of_exp
tff(fact_1837_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_of_2
tff(fact_1838_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),P2: product_prod(B,C)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),P2)))
     => ~ ! [X4: B,Y2: C] :
            ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y2) )
           => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(C,set(A),aa(B,fun(C,set(A)),C2,X4),Y2))) ) ) ).

% mem_case_prodE
tff(fact_1839_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,bool)),P2: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C2),P2))
     => ~ ! [X4: A,Y2: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2) )
           => ~ pp(aa(B,bool,aa(A,fun(B,bool),C2,X4),Y2)) ) ) ).

% case_prodE
tff(fact_1840_case__prodD,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,bool)),A2: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)))
     => pp(aa(B,bool,aa(A,fun(B,bool),F2,A2),B2)) ) ).

% case_prodD
tff(fact_1841_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,bool))),P2: product_prod(A,B),Z: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C2),P2),Z))
     => ~ ! [X4: A,Y2: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2) )
           => ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,X4),Y2),Z)) ) ) ).

% case_prodE'
tff(fact_1842_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,bool))),A2: A,B2: B,C2: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C2))
     => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R,A2),B2),C2)) ) ).

% case_prodD'
tff(fact_1843_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P))) ) ).

% zero_less_eq_of_bool
tff(fact_1844_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A))) ) ).

% of_bool_less_eq_one
tff(fact_1845_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ~ ( ( pp(P2)
                & ~ pp(aa(A,bool,P,one_one(A))) )
              | ( ~ pp(P2)
                & ~ pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool_asm
tff(fact_1846_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ( ( pp(P2)
             => pp(aa(A,bool,P,one_one(A))) )
            & ( ~ pp(P2)
             => pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool
tff(fact_1847_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool] :
          ( ( pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = one_one(A) ) )
          & ( ~ pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = zero_zero(A) ) ) ) ) ).

% of_bool_def
tff(fact_1848_finite__set__decode,axiom,
    ! [N: nat] : pp(aa(set(nat),bool,finite_finite2(nat),nat_set_decode(N))) ).

% finite_set_decode
tff(fact_1849_set__encode__eq,axiom,
    ! [A3: set(nat),B3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( pp(aa(set(nat),bool,finite_finite2(nat),B3))
       => ( ( aa(set(nat),nat,nat_set_encode,A3) = aa(set(nat),nat,nat_set_encode,B3) )
        <=> ( A3 = B3 ) ) ) ) ).

% set_encode_eq
tff(fact_1850_set__encode__inf,axiom,
    ! [A3: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( aa(set(nat),nat,nat_set_encode,A3) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_1851_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% subset_decode_imp_le
tff(fact_1852_divmod__int__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(int,M,N) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N))) ).

% divmod_int_def
tff(fact_1853_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),divide_divide(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_1854_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(nat,M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N))) ).

% divmod'_nat_def
tff(fact_1855_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ).

% exp_mod_exp
tff(fact_1856_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,M,N)),modulo_modulo(nat,M,N)) ).

% divmod_nat_def
tff(fact_1857_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] : divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ).

% exp_div_exp_eq
tff(fact_1858_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M,aa(num,num,bit0,N))) ) ) ) ) ).

% divmod_divmod_step
tff(fact_1859_listrel1p__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( listrel1p(A,R2,Xs,Ys)
    <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R2))))) ) ).

% listrel1p_def
tff(fact_1860_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),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_bk(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_1861_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q4),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),R2),zero_zero(int))))) ).

% Divides.adjust_div_eq
tff(fact_1862_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_bl(int,fun(int,fun(int,bool)),D2))) ).

% int_ge_less_than2_def
tff(fact_1863_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_bm(int,fun(int,fun(int,bool)),D2))) ).

% int_ge_less_than_def
tff(fact_1864_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_1865_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_1866_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A2)) = A2 ) ).

% add.inverse_inverse
tff(fact_1867_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_1868_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% compl_le_compl_iff
tff(fact_1869_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% neg_le_iff_le
tff(fact_1870_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = A2 )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_1871_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_1872_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_1873_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A2) )
        <=> ( zero_zero(A) = A2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_1874_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_1875_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% compl_less_compl_iff
tff(fact_1876_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% neg_less_iff_less
tff(fact_1877_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_1878_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_1879_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_1880_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% minus_diff_eq
tff(fact_1881_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(80)
tff(fact_1882_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% neg_0_le_iff_le
tff(fact_1883_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% neg_le_0_iff_le
tff(fact_1884_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% less_eq_neg_nonpos
tff(fact_1885_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% neg_less_eq_nonneg
tff(fact_1886_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% neg_less_0_iff_less
tff(fact_1887_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% neg_0_less_iff_less
tff(fact_1888_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% neg_less_pos
tff(fact_1889_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% less_neg_neg
tff(fact_1890_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),A2)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_1891_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_1892_verit__minus__simplify_I3_J,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),zero_zero(B)),B2) = aa(B,B,uminus_uminus(B),B2) ) ).

% verit_minus_simplify(3)
tff(fact_1893_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = aa(A,A,uminus_uminus(A),A2) ) ).

% diff_0
tff(fact_1894_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_1895_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) ) ).

% diff_minus_eq_add
tff(fact_1896_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(81)
tff(fact_1897_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),aa(num,num,bit1,N))) ).

% semiring_norm(77)
tff(fact_1898_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_1899_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_1900_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% diff_numeral_special(12)
tff(fact_1901_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : modulo_modulo(A,A2,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_1902_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(4)
tff(fact_1903_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(3)
tff(fact_1904_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(2)
tff(fact_1905_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M)) ) ) ).

% neg_numeral_le_iff
tff(fact_1906_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M)) ) ) ).

% neg_numeral_less_iff
tff(fact_1907_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(74)
tff(fact_1908_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(79)
tff(fact_1909_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_1910_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_1911_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_1912_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A2: A] :
          ( ( divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = A2 )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_1913_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W2: num] :
          ( ( A2 = divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) )
        <=> ( ( ( 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),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = B2 ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_1914_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_1915_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_1916_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_1917_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_1918_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_1919_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% equation_minus_iff
tff(fact_1920_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A2 ) ) ) ).

% minus_equation_iff
tff(fact_1921_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_le_swap2
tff(fact_1922_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_le_swap1
tff(fact_1923_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X))) ) ) ).

% compl_mono
tff(fact_1924_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% le_imp_neg_le
tff(fact_1925_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A2)) ) ) ).

% minus_le_iff
tff(fact_1926_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% le_minus_iff
tff(fact_1927_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_less_swap1
tff(fact_1928_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_less_swap2
tff(fact_1929_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A2)) ) ) ).

% minus_less_iff
tff(fact_1930_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% less_minus_iff
tff(fact_1931_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% verit_negate_coefficient(2)
tff(fact_1932_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_1933_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% add.inverse_distrib_swap
tff(fact_1934_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_diff_commute
tff(fact_1935_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_le_neg_numeral
tff(fact_1936_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_le_numeral
tff(fact_1937_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% zero_neq_neg_numeral
tff(fact_1938_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_less_neg_numeral
tff(fact_1939_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_less_numeral
tff(fact_1940_xor__num_Ocases,axiom,
    ! [X: product_prod(num,num)] :
      ( ( X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N2))
       => ( ! [N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N2))
         => ( ! [M5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one2)
           => ( ! [M5: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N2))
             => ( ! [M5: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N2))
               => ( ! [M5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one2)
                 => ( ! [M5: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N2))
                   => ~ ! [M5: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_1941_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% le_minus_one_simps(2)
tff(fact_1942_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(4)
tff(fact_1943_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% add_eq_0_iff
tff(fact_1944_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_1945_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A2) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_1946_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_1947_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_1948_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_1949_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% less_minus_one_simps(2)
tff(fact_1950_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(4)
tff(fact_1951_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_1952_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_1953_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1954_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_1955_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B3: A,K: A,B2: A,A2: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_1956_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_le_zero
tff(fact_1957_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_le_neg_numeral
tff(fact_1958_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_less_neg_numeral
tff(fact_1959_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_less_zero
tff(fact_1960_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(3)
tff(fact_1961_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% le_minus_one_simps(1)
tff(fact_1962_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% less_minus_one_simps(1)
tff(fact_1963_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(3)
tff(fact_1964_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_le_one
tff(fact_1965_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_le_numeral
tff(fact_1966_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% neg_numeral_le_neg_one
tff(fact_1967_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_le_neg_one
tff(fact_1968_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_le_neg_numeral
tff(fact_1969_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_1970_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_less_neg_numeral
tff(fact_1971_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_less_neg_one
tff(fact_1972_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_less_numeral
tff(fact_1973_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_less_one
tff(fact_1974_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,uminus_uminus(A),B2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_1975_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)) = A2 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_1976_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_1977_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_1978_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( divide_divide(A,A2,B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_1979_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_1980_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_1981_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_1982_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_1983_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_1984_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_1985_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( divide_divide(A,B2,C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_1986_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,B2,C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_1987_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q4: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q4))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_1988_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A2,Z))),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A2,Z))),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1989_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,X,Z))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1990_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A2,Z))),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A2,Z))),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_1991_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,Z)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,Z)),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_1992_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,X,Z))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1993_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_1994_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),B3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),N)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,B3,N)),divide_divide(int,A3,N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_1995_div__eq__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_1996_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_1997_num_Osize_I6_J,axiom,
    ! [X32: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_1998_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_1999_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_2000_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2001_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2002_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2003_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2004_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2005_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2006_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q4: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q4))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q4))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2007_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q4: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q4))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q4))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2008_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2009_zmod__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2010_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q4: int,R2: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R2),zero_zero(int)),aa(int,int,uminus_uminus(int),Q4),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q4)),one_one(int)))),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R2),zero_zero(int)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_2011_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2012_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2013_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))) ) ) ) ).

% square_le_1
tff(fact_2014_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
       => ( divide_divide(int,K,L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_2015_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% odd_mod_4_div_2
tff(fact_2016_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_2017_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% signed_take_bit_rec
tff(fact_2018_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),aa(num,num,bit0,one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_2019_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ( ( ( K = zero_zero(int) )
           => ( aa(int,A,ring_1_of_int(A),K) = zero_zero(A) ) )
          & ( ( K != zero_zero(int) )
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
               => ( aa(int,A,ring_1_of_int(A),K) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K))) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
               => ( aa(int,A,ring_1_of_int(A),K) = if(A,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(int,A,ring_1_of_int(A),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(int,A,ring_1_of_int(A),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),one_one(A))) ) ) ) ) ) ) ).

% of_int_code_if
tff(fact_2020_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A2) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% signed_take_bit_0
tff(fact_2021_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2022_Compl__anti__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B3)),aa(set(A),set(A),uminus_uminus(set(A)),A3))) ) ).

% Compl_anti_mono
tff(fact_2023_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B3)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ).

% Compl_subset_Compl_iff
tff(fact_2024_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2025_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_2026_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_2027_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_2028_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z)) ) ) ).

% of_int_le_iff
tff(fact_2029_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z)) ) ) ).

% of_int_less_iff
tff(fact_2030_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int))) ) ) ).

% of_int_le_0_iff
tff(fact_2031_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) ) ) ).

% of_int_0_le_iff
tff(fact_2032_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ) ).

% of_int_0_less_iff
tff(fact_2033_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ) ).

% of_int_less_0_iff
tff(fact_2034_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_le_numeral_iff
tff(fact_2035_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_le_iff
tff(fact_2036_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_less_numeral_iff
tff(fact_2037_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_less_iff
tff(fact_2038_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),one_one(int))) ) ) ).

% of_int_le_1_iff
tff(fact_2039_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z)) ) ) ).

% of_int_1_le_iff
tff(fact_2040_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),one_one(int))) ) ) ).

% of_int_less_1_iff
tff(fact_2041_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z)) ) ) ).

% of_int_1_less_iff
tff(fact_2042_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)),X)) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2043_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2))) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2044_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)),X)) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2045_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2))) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2046_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2047_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2048_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2049_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2050_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A2)) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2051_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2052_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A2)) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2053_subset__Compl__self__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_2054_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(real,real,uminus_uminus(real),X))) ) ).

% real_add_less_0_iff
tff(fact_2055_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),X)),Y)) ) ).

% real_0_less_add_iff
tff(fact_2056_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,if(fun(A,A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M),bit_se2584673776208193580ke_bit(A,N),bit_ri4674362597316999326ke_bit(A,M)),A2) ) ).

% signed_take_bit_take_bit
tff(fact_2057_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),M))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(int,real,ring_1_of_int(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),M)),one_one(real)))) ) ).

% int_le_real_less
tff(fact_2058_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),M))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))),aa(int,real,ring_1_of_int(real),M))) ) ).

% int_less_real_le
tff(fact_2059_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% of_int_nonneg
tff(fact_2060_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% of_int_pos
tff(fact_2061_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2062_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_2063_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2064_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2065_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2066_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2067_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2068_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2069_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X4)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),one_one(int)))))
          & ! [Y3: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y3)),X))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y3),one_one(int))))) )
             => ( Y3 = X4 ) ) ) ) ).

% floor_exists1
tff(fact_2070_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int))))) ) ) ).

% floor_exists
tff(fact_2071_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),Y)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))
           => ( archimedean_round(A,X) = Y ) ) ) ) ).

% round_unique
tff(fact_2072_set__encode__insert,axiom,
    ! [A3: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
     => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),A3))
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),N),A3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_2073_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_bn(nat,fun(nat,fun(nat,nat)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% arith_series_nat
tff(fact_2074_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ).

% Suc_0_xor_eq
tff(fact_2075_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ).

% xor_Suc_0_eq
tff(fact_2076_insertCI,axiom,
    ! [A: $tType,A2: A,B3: set(A),B2: A] :
      ( ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
       => ( A2 = B2 ) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3))) ) ).

% insertCI
tff(fact_2077_insert__iff,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),A3)))
    <=> ( ( A2 = B2 )
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ).

% insert_iff
tff(fact_2078_insert__absorb2,axiom,
    ! [A: $tType,X: A,A3: set(A)] : 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),X),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3) ).

% insert_absorb2
tff(fact_2079_ComplI,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),uminus_uminus(set(A)),A3))) ) ).

% ComplI
tff(fact_2080_Compl__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),uminus_uminus(set(A)),A3)))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) ) ).

% Compl_iff
tff(fact_2081_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),uminus_uminus(set(A)),B3) )
    <=> ( A3 = B3 ) ) ).

% Compl_eq_Compl_iff
tff(fact_2082_singletonI,axiom,
    ! [A: $tType,A2: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) ).

% singletonI
tff(fact_2083_finite__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)))
    <=> pp(aa(set(A),bool,finite_finite2(A),A3)) ) ).

% finite_insert
tff(fact_2084_insert__subset,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),B3))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ).

% insert_subset
tff(fact_2085_Diff__insert0,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ) ) ).

% Diff_insert0
tff(fact_2086_insert__Diff1,axiom,
    ! [A: $tType,X: A,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ) ) ).

% insert_Diff1
tff(fact_2087_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),zero_zero(A)) = A2 ) ).

% xor.right_neutral
tff(fact_2088_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A2) = A2 ) ).

% xor.left_neutral
tff(fact_2089_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),A2) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_2090_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),X) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_2091_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_bo(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2092_singleton__conv,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bp(A,fun(A,bool),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_2093_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),fequal(A),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_2094_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_2095_sum__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [F4: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),F4))
         => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),F4) = zero_zero(A) )
          <=> ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),F4))
               => ( aa(B,A,F2,X3) = zero_zero(A) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_2096_sum_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ~ pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = zero_zero(A) ) ) ) ).

% sum.infinite
tff(fact_2097_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) )
    <=> ( ( A2 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2098_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2099_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),bot_bot(set(A))) )
        <=> ( ( A2 = B2 )
            & ( B2 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_2100_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : set_or1337092689740270186AtMost(A,A2,A2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_2101_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) ).

% insert_Diff_single
tff(fact_2102_finite__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3))))
    <=> pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ) ).

% finite_Diff_insert
tff(fact_2103_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2104_sum_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_bq(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_bq(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = zero_zero(A) ) ) ) ) ) ).

% sum.delta
tff(fact_2105_sum_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_br(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_br(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = zero_zero(A) ) ) ) ) ) ).

% sum.delta'
tff(fact_2106_sum_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_2107_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3)) ) ).

% subset_Compl_singleton
tff(fact_2108_set__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_2109_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_2110_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_2111_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_2112_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_2113_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_2114_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: set(nat),C2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bs(fun(nat,A),fun(nat,A),C2)),A3) = aa(nat,A,C2,zero_zero(nat)) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bs(fun(nat,A),fun(nat,A),C2)),A3) = zero_zero(A) ) ) ) ) ).

% sum_zero_power
tff(fact_2115_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: set(nat),C2: fun(nat,A),D2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = divide_divide(A,aa(nat,A,C2,zero_zero(nat)),aa(nat,A,D2,zero_zero(nat))) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = zero_zero(A) ) ) ) ) ).

% sum_zero_power'
tff(fact_2116_uminus__set__def,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),uminus_uminus(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3))) ).

% uminus_set_def
tff(fact_2117_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bu(fun(A,bool),fun(A,bool),P)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,bool),set(A),collect(A),P)) ).

% Collect_neg_eq
tff(fact_2118_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bv(set(A),fun(A,bool),A3)) ).

% Compl_eq
tff(fact_2119_ComplD,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),uminus_uminus(set(A)),A3)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) ) ).

% ComplD
tff(fact_2120_double__complement,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = A3 ).

% double_complement
tff(fact_2121_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: fun(A,bool)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(fun(A,bool),set(A),collect(A),P)) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_bw(A,fun(fun(A,bool),fun(A,bool)),A2),P)) ).

% insert_Collect
tff(fact_2122_insert__compr,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_bx(A,fun(set(A),fun(A,bool)),A2),B3)) ).

% insert_compr
tff(fact_2123_insertE,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),A3)))
     => ( ( A2 != B2 )
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ).

% insertE
tff(fact_2124_insertI1,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3))) ).

% insertI1
tff(fact_2125_insertI2,axiom,
    ! [A: $tType,A2: A,B3: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3))) ) ).

% insertI2
tff(fact_2126_Set_Oset__insert,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ~ ! [B8: set(A)] :
            ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B8) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B8)) ) ) ).

% Set.set_insert
tff(fact_2127_insert__ident,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3) )
        <=> ( A3 = B3 ) ) ) ) ).

% insert_ident
tff(fact_2128_insert__absorb,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) = A3 ) ) ).

% insert_absorb
tff(fact_2129_insert__eq__iff,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A,B3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),B3))
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3) )
        <=> ( ( ( A2 = B2 )
             => ( A3 = B3 ) )
            & ( ( A2 != B2 )
             => ? [C6: set(A)] :
                  ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),C6) )
                  & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),C6))
                  & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),C6) )
                  & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C6)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_2130_insert__commute,axiom,
    ! [A: $tType,X: A,Y: A,A3: set(A)] : 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),Y),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) ).

% insert_commute
tff(fact_2131_mk__disjoint__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ? [B8: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B8) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B8)) ) ) ).

% mk_disjoint_insert
tff(fact_2132_sum_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)) ) ) ) ) ) ).

% sum.insert_if
tff(fact_2133_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
               => ( aa(B,A,G,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_2134_sum_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => ( aa(B,A,G,X4) = zero_zero(A) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = zero_zero(A) ) ) ) ).

% sum.neutral
tff(fact_2135_singleton__inject,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_2136_insert__not__empty,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_2137_doubleton__eq__iff,axiom,
    ! [A: $tType,A2: A,B2: A,C2: A,D2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),D2),bot_bot(set(A)))) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2138_singleton__iff,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_2139_singletonD,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_2140_finite_OinsertI,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3))) ) ).

% finite.insertI
tff(fact_2141_subset__insertI2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3))) ) ).

% subset_insertI2
tff(fact_2142_subset__insertI,axiom,
    ! [A: $tType,B3: set(A),A2: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3))) ).

% subset_insertI
tff(fact_2143_subset__insert,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ).

% subset_insert
tff(fact_2144_insert__mono,axiom,
    ! [A: $tType,C3: set(A),D6: set(A),A2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),D6))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),C3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),D6))) ) ).

% insert_mono
tff(fact_2145_insert__subsetI,axiom,
    ! [A: $tType,X: A,A3: set(A),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),A3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),X6)),A3)) ) ) ).

% insert_subsetI
tff(fact_2146_insert__Diff__if,axiom,
    ! [A: $tType,X: A,B3: set(A),A3: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) ) ) ) ).

% insert_Diff_if
tff(fact_2147_sum__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K4: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),K4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),aa(B,A,G,I2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),K4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),K4))) ) ) ).

% sum_mono
tff(fact_2148_sum_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: set(C),G: fun(B,fun(C,A)),R: fun(B,fun(C,bool))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(C),bool,finite_finite2(C),B3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_bz(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B3),G),R)),A3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_cc(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A3),G),R)),B3) ) ) ) ) ).

% sum.swap_restrict
tff(fact_2149_sum__diff1__nat,axiom,
    ! [A: $tType,A2: A,A3: set(A),F2: fun(A,nat)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(A,nat,F2,A2)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) ) ) ) ).

% sum_diff1_nat
tff(fact_2150_Collect__conv__if,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( ( pp(aa(A,bool,P,A2))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_cd(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P,A2))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_cd(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if
tff(fact_2151_Collect__conv__if2,axiom,
    ! [A: $tType,P: fun(A,bool),A2: A] :
      ( ( pp(aa(A,bool,P,A2))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_ce(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P,A2))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_ce(fun(A,bool),fun(A,fun(A,bool)),P),A2)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if2
tff(fact_2152_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y))) ) ) ).

% round_mono
tff(fact_2153_sum_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ).

% sum.insert_remove
tff(fact_2154_sum_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% sum.remove
tff(fact_2155_sum__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A3: set(B),A2: B,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(B,A,F2,A2)) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3) ) ) ) ) ) ).

% sum_diff1
tff(fact_2156_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3))) ) ) ).

% sum_nonneg
tff(fact_2157_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),zero_zero(A))) ) ) ).

% sum_nonpos
tff(fact_2158_sum__mono__inv,axiom,
    ! [A: $tType,I7: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F2: fun(I7,A),I6: set(I7),G: fun(I7,A),I: I7] :
          ( ( aa(set(I7),A,aa(fun(I7,A),fun(set(I7),A),groups7311177749621191930dd_sum(I7,A),F2),I6) = aa(set(I7),A,aa(fun(I7,A),fun(set(I7),A),groups7311177749621191930dd_sum(I7,A),G),I6) )
         => ( ! [I2: I7] :
                ( pp(aa(set(I7),bool,aa(I7,fun(set(I7),bool),member(I7),I2),I6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I7,A,F2,I2)),aa(I7,A,G,I2))) )
           => ( pp(aa(set(I7),bool,aa(I7,fun(set(I7),bool),member(I7),I),I6))
             => ( pp(aa(set(I7),bool,finite_finite2(I7),I6))
               => ( aa(I7,A,F2,I) = aa(I7,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_2159_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
         => ( ! [X4: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,X4)),A3))
               => ( aa(nat,A,F2,aa(nat,nat,suc,X4)) = aa(nat,A,G,aa(nat,nat,suc,X4)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_2160_sum_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_cf(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,B2,A2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_cf(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))) ) ) ) ) ) ).

% sum.delta_remove
tff(fact_2161_infinite__finite__induct,axiom,
    ! [A: $tType,P: fun(set(A),bool),A3: set(A)] :
      ( ! [A5: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => pp(aa(set(A),bool,P,A5)) )
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [X4: A,F5: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),F5))
             => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),F5))
               => ( pp(aa(set(A),bool,P,F5))
                 => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),F5))) ) ) )
         => pp(aa(set(A),bool,P,A3)) ) ) ) ).

% infinite_finite_induct
tff(fact_2162_finite__ne__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( ( F4 != bot_bot(set(A)) )
       => ( ! [X4: A] : pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A)))))
         => ( ! [X4: A,F5: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F5))
               => ( ( F5 != bot_bot(set(A)) )
                 => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),F5))
                   => ( pp(aa(set(A),bool,P,F5))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),F5))) ) ) ) )
           => pp(aa(set(A),bool,P,F4)) ) ) ) ) ).

% finite_ne_induct
tff(fact_2163_finite__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [X4: A,F5: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),F5))
             => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),F5))
               => ( pp(aa(set(A),bool,P,F5))
                 => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),F5))) ) ) )
         => pp(aa(set(A),bool,P,F4)) ) ) ) ).

% finite_induct
tff(fact_2164_finite_Osimps,axiom,
    ! [A: $tType,A2: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A2))
    <=> ( ( A2 = bot_bot(set(A)) )
        | ? [A8: set(A),A7: A] :
            ( ( A2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),A8) )
            & pp(aa(set(A),bool,finite_finite2(A),A8)) ) ) ) ).

% finite.simps
tff(fact_2165_finite_Ocases,axiom,
    ! [A: $tType,A2: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A2))
     => ( ( A2 != bot_bot(set(A)) )
       => ~ ! [A5: set(A)] :
              ( ? [A4: A] : A2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),A5)
             => ~ pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ) ).

% finite.cases
tff(fact_2166_subset__singletonD,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
     => ( ( A3 = bot_bot(set(A)) )
        | ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_2167_subset__singleton__iff,axiom,
    ! [A: $tType,X6: set(A),A2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))
    <=> ( ( X6 = bot_bot(set(A)) )
        | ( X6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_2168_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_2169_Diff__insert__absorb,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_2170_Diff__insert2,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))),B3) ).

% Diff_insert2
tff(fact_2171_insert__Diff,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = A3 ) ) ).

% insert_Diff
tff(fact_2172_Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_2173_Compl__insert,axiom,
    ! [A: $tType,X: A,A3: 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),X),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_2174_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),X: A,C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),C3))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C3)))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_2175_sum_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_cg(set(B),fun(fun(B,bool),fun(B,bool)),A3),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_ch(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_2176_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: C,A3: set(C),F2: fun(C,B)] :
          ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I),A3))
         => ( ! [X4: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),A3),aa(set(C),set(C),aa(C,fun(set(C),set(C)),insert2(C),I),bot_bot(set(C))))))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(C,B,F2,X4))) )
           => ( pp(aa(set(C),bool,finite_finite2(C),A3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F2,I)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F2),A3))) ) ) ) ) ).

% member_le_sum
tff(fact_2177_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,G,X4)),aa(A,nat,F2,X4))) )
     => ( 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_ci(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_2178_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),T2: set(C),G: fun(C,A),I: fun(C,B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(C),bool,finite_finite2(C),T2))
           => ( ! [X4: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),T2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(C,A,G,X4))) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S))
                   => ? [Xa: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Xa),T2))
                        & ( aa(C,B,I,Xa) = X4 )
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(C,A,G,Xa))) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),S)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T2))) ) ) ) ) ) ).

% sum_le_included
tff(fact_2179_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X4))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3) = zero_zero(A) )
            <=> ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                 => ( aa(B,A,F2,X3) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_2180_sum__strict__mono__ex1,axiom,
    ! [A: $tType,I7: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: set(I7),F2: fun(I7,A),G: fun(I7,A)] :
          ( pp(aa(set(I7),bool,finite_finite2(I7),A3))
         => ( ! [X4: I7] :
                ( pp(aa(set(I7),bool,aa(I7,fun(set(I7),bool),member(I7),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I7,A,F2,X4)),aa(I7,A,G,X4))) )
           => ( ? [X2: I7] :
                  ( pp(aa(set(I7),bool,aa(I7,fun(set(I7),bool),member(I7),X2),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(I7,A,F2,X2)),aa(I7,A,G,X2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(I7),A,aa(fun(I7,A),fun(set(I7),A),groups7311177749621191930dd_sum(I7,A),F2),A3)),aa(set(I7),A,aa(fun(I7,A),fun(set(I7),A),groups7311177749621191930dd_sum(I7,A),G),A3))) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_2181_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,bool)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R,zero_zero(A)),zero_zero(A)))
         => ( ! [X15: A,Y15: A,X23: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X15),X23))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R,Y15),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(A,A,aa(A,fun(A,A),plus_plus(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y23))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S2))
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H,X4)),aa(B,A,G,X4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2))) ) ) ) ) ) ).

% sum.related
tff(fact_2182_sum__strict__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( A3 != bot_bot(set(B)) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X4)),aa(B,A,G,X4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3))) ) ) ) ) ).

% sum_strict_mono
tff(fact_2183_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
            & ( aa(A,nat,F2,X3) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_2184_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S5: set(B),T5: set(C),S2: set(B),I: fun(C,B),J: fun(B,C),T3: set(C),G: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T5))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
                 => ( aa(C,B,I,aa(B,C,J,A4)) = A4 ) )
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J,A4)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T3),T5))) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T3),T5)))
                     => ( aa(B,C,J,aa(C,B,I,B4)) = B4 ) )
                 => ( ! [B4: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T3),T5)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I,B4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5))) )
                   => ( ! [A4: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S5))
                         => ( aa(B,A,G,A4) = zero_zero(A) ) )
                     => ( ! [B4: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T5))
                           => ( aa(C,A,H,B4) = zero_zero(A) ) )
                       => ( ! [A4: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S2))
                             => ( aa(C,A,H,aa(B,C,J,A4)) = aa(B,A,G,A4) ) )
                         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),H),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_2185_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),N: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = aa(nat,nat,suc,N) )
     => ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X4))) ) ) ).

% sum_SucD
tff(fact_2186_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = one_one(nat) )
      <=> ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
            & ( aa(A,nat,F2,X3) = one_one(nat) )
            & ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A3))
               => ( ( X3 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2187_finite__ranking__induct,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [S2: set(B),P: fun(set(B),bool),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( pp(aa(set(B),bool,P,bot_bot(set(B))))
           => ( ! [X4: B,S6: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite2(B),S6))
                 => ( ! [Y3: B] :
                        ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),S6))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,Y3)),aa(B,A,F2,X4))) )
                   => ( pp(aa(set(B),bool,P,S6))
                     => pp(aa(set(B),bool,P,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X4),S6))) ) ) )
             => pp(aa(set(B),bool,P,S2)) ) ) ) ) ).

% finite_ranking_induct
tff(fact_2188_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
           => ( ! [B4: A,A5: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite2(A),A5))
                 => ( ! [X2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A5))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),X2)) )
                   => ( pp(aa(set(A),bool,P,A5))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),A5))) ) ) )
             => pp(aa(set(A),bool,P,A3)) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2189_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
           => ( ! [B4: A,A5: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite2(A),A5))
                 => ( ! [X2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A5))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),B4)) )
                   => ( pp(aa(set(A),bool,P,A5))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),A5))) ) ) )
             => pp(aa(set(A),bool,P,A3)) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2190_finite__subset__induct,axiom,
    ! [A: $tType,F4: set(A),A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F4),A3))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A4: A,F5: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F5))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),F5))
                   => ( pp(aa(set(A),bool,P,F5))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),F5))) ) ) ) )
           => pp(aa(set(A),bool,P,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_2191_finite__subset__induct_H,axiom,
    ! [A: $tType,F4: set(A),A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F4),A3))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A4: A,F5: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F5))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F5),A3))
                   => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),F5))
                     => ( pp(aa(set(A),bool,P,F5))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),F5))) ) ) ) ) )
           => pp(aa(set(A),bool,P,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2192_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),F2: fun(B,A),B3: A,I: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I2))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),S) = B3 )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),B3)) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2193_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),F2: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I2))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),S) = zero_zero(A) )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),S))
               => ( aa(B,A,F2,I) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2194_finite__empty__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,P,A3))
       => ( ! [A4: A,A5: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A5))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A5))
               => ( pp(aa(set(A),bool,P,A5))
                 => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))))) ) ) )
         => pp(aa(set(A),bool,P,bot_bot(set(A)))) ) ) ) ).

% finite_empty_induct
tff(fact_2195_infinite__coinduct,axiom,
    ! [A: $tType,X6: fun(set(A),bool),A3: set(A)] :
      ( pp(aa(set(A),bool,X6,A3))
     => ( ! [A5: set(A)] :
            ( pp(aa(set(A),bool,X6,A5))
           => ? [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A5))
                & ( pp(aa(set(A),bool,X6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))))
                  | ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A)))))) ) ) )
       => ~ pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% infinite_coinduct
tff(fact_2196_infinite__remove,axiom,
    ! [A: $tType,S2: set(A),A2: A] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
     => ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ).

% infinite_remove
tff(fact_2197_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),X: A,B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3))) ) ).

% Diff_single_insert
tff(fact_2198_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ) ).

% subset_insert_iff
tff(fact_2199_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_2200_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( set_or1337092689740270186AtMost(nat,M,N) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_2201_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_2202_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = set_or1337092689740270186AtMost(nat,M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_2203_sum_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_cj(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2204_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs)))) ).

% set_update_subset_insert
tff(fact_2205_sum__pos2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I6: set(B),I: B,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F2,I)))
             => ( ! [I2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I2))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),I6))) ) ) ) ) ) ).

% sum_pos2
tff(fact_2206_sum__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I6: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( ( I6 != bot_bot(set(B)) )
           => ( ! [I2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F2,I2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),I6))) ) ) ) ) ).

% sum_pos
tff(fact_2207_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T3: set(B),S2: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,G,X4) = zero_zero(A) ) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_2208_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T3: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,H,X4) = zero_zero(A) ) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_2209_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T3: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,G,X4) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_2210_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T3: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,G,X4) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T3) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_2211_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C3: set(B),A3: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),A3)))
                   => ( aa(B,A,G,A4) = zero_zero(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B3)))
                     => ( aa(B,A,H,B4) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),C3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C3) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B3) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_2212_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C3: set(B),A3: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),A3)))
                   => ( aa(B,A,G,A4) = zero_zero(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B3)))
                     => ( aa(B,A,H,B4) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B3) )
                  <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),C3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C3) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_2213_sum_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B3: set(B),A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B3)) ) ) ) ) ).

% sum.subset_diff
tff(fact_2214_sum__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A3: set(B),B3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B3)) ) ) ) ) ).

% sum_diff
tff(fact_2215_sum__diff__nat,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B3)) ) ) ) ).

% sum_diff_nat
tff(fact_2216_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_2217_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2218_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2219_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2220_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),bool),B3: set(A)] :
      ( pp(aa(set(A),bool,P,bot_bot(set(A))))
     => ( ( ~ pp(aa(set(A),bool,finite_finite2(A),B3))
         => pp(aa(set(A),bool,P,B3)) )
       => ( ! [A5: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A5))
             => ( ( A5 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B3))
                 => ( ! [X2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A5))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A5)) ) ) ) )
         => pp(aa(set(A),bool,P,B3)) ) ) ) ).

% remove_induct
tff(fact_2221_finite__remove__induct,axiom,
    ! [A: $tType,B3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [A5: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A5))
             => ( ( A5 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B3))
                 => ( ! [X2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A5))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A5)) ) ) ) )
         => pp(aa(set(A),bool,P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_2222_finite__induct__select,axiom,
    ! [A: $tType,S2: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [T6: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),T6),S2))
             => ( pp(aa(set(A),bool,P,T6))
               => ? [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T6)))
                    & pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),T6))) ) ) )
         => pp(aa(set(A),bool,P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_2223_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),B3)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3)) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_2224_set__replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,suc,N)),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_2225_set__replicate__conv__if,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = bot_bot(set(A)) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ).

% set_replicate_conv_if
tff(fact_2226_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2227_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cl(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_2228_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [B3: set(B),A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
           => ( ! [B4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B3),A3)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,B4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B3))) ) ) ) ) ).

% sum_mono2
tff(fact_2229_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2230_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X6) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_2231_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B3: set(A),A3: set(A),B2: A,F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B3))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))
             => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),zero_zero(B)),aa(A,B,F2,B2)))
               => ( ! [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X4))) )
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B3))) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_2232_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cm(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,M)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cm(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) ) ) ) ).

% sum_natinterval_diff
tff(fact_2233_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cn(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,N)),aa(nat,A,F2,M)) ) ) ) ).

% sum_telescope''
tff(fact_2234_exists__least__lemma,axiom,
    ! [P: fun(nat,bool)] :
      ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
       => ? [N2: nat] :
            ( ~ pp(aa(nat,bool,P,N2))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_2235_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N))) ) ).

% mask_eq_sum_exp
tff(fact_2236_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).

% ex_le_of_int
tff(fact_2237_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).

% ex_less_of_int
tff(fact_2238_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),X)) ) ).

% ex_of_int_less
tff(fact_2239_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_2240_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% of_int_round_le
tff(fact_2241_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_ge
tff(fact_2242_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_gt
tff(fact_2243_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_2244_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N))) ).

% mask_eq_sum_exp_nat
tff(fact_2245_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_co(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% gauss_sum_nat
tff(fact_2246_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [N: nat,M: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( ( ( X = one_one(A) )
               => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M)) ) )
              & ( ( X != one_one(A) )
               => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ) ) ).

% sum_gp
tff(fact_2247_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_2248_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_2249_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum
tff(fact_2250_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D2: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_cp(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% arith_series
tff(fact_2251_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_2252_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_cq(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D2))) ) ).

% double_arith_series
tff(fact_2253_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( M = N ) ) ) ).

% of_nat_eq_iff
tff(fact_2254_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_2255_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int)))
    <=> ~ ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% xor_negative_int_iff
tff(fact_2256_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_2257_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( zero_zero(nat) = N ) ) ) ).

% of_nat_0_eq_iff
tff(fact_2258_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_2259_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_iff
tff(fact_2260_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% of_nat_le_iff
tff(fact_2261_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(num,real,numeral_numeral(real),W2)),aa(nat,real,semiring_1_of_nat(real),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),W2)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_2262_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W2: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(num,real,numeral_numeral(real),W2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),W2))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_2263_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_add
tff(fact_2264_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),N)),aa(nat,real,semiring_1_of_nat(real),M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_2265_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_2266_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_2267_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_2268_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_2269_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zless
tff(fact_2270_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: bool] : aa(nat,A,semiring_1_of_nat(A),aa(bool,nat,zero_neq_one_of_bool(nat),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% of_nat_of_bool
tff(fact_2271_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_2272_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_2273_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% of_nat_0_less_iff
tff(fact_2274_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)),X)) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_2275_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2))) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_2276_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)),X)) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2277_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2))) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2278_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),X)),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2279_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N))) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2280_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,N: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X)) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2281_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,N: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X)) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2282_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N))) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2283_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% real_arch_simple
tff(fact_2284_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% reals_Archimedean2
tff(fact_2285_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_2286_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),X)) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2287_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% of_nat_0_le_iff
tff(fact_2288_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))) ) ).

% of_nat_less_0_iff
tff(fact_2289_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_2290_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) ) ) ).

% less_imp_of_nat_less
tff(fact_2291_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_imp_less
tff(fact_2292_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J))) ) ) ).

% of_nat_mono
tff(fact_2293_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_2294_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(2)
tff(fact_2295_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(3)
tff(fact_2296_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% zle_int
tff(fact_2297_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% not_int_zless_negative
tff(fact_2298_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_2299_nat__less__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),Xa))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_less_as_int
tff(fact_2300_nat__leq__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Xa))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_leq_as_int
tff(fact_2301_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),X))) ) ) ).

% ex_less_of_nat_mult
tff(fact_2302_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).

% of_nat_diff
tff(fact_2303_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ! [Y3: real] :
        ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),X))) ) ).

% reals_Archimedean3
tff(fact_2304_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ) ) ).

% int_cases4
tff(fact_2305_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z))
    <=> ? [N3: 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,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_2306_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))))
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_2307_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_cr(complex,complex)),aa(fun(complex,bool),set(complex),collect(complex),aTP_Lamp_cs(nat,fun(complex,bool),N))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_2308_sum__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_cr(complex,complex)),aa(fun(complex,bool),set(complex),collect(complex),aa(complex,fun(complex,bool),aTP_Lamp_av(nat,fun(complex,fun(complex,bool)),N),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_2309_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ~ ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% pos_int_cases
tff(fact_2310_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ? [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
          & ( K = aa(nat,int,semiring_1_of_nat(int),N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2311_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
             => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ) ).

% int_cases3
tff(fact_2312_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),N)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% nat_less_real_le
tff(fact_2313_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),M)),one_one(real)))) ) ).

% nat_le_real_less
tff(fact_2314_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_2315_negD,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
     => ? [N2: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% negD
tff(fact_2316_negative__zless__0,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int))) ).

% negative_zless_0
tff(fact_2317_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = zero_zero(int) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ) ) ).

% int_ops(6)
tff(fact_2318_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
       => ( set_or1337092689740270186AtMost(int,I,J) = bot_bot(set(int)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
       => ( set_or1337092689740270186AtMost(int,I,J) = 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_2319_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E3))
         => ~ ! [N2: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),E3)) ) ) ).

% nat_approx_posE
tff(fact_2320_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))) ) ).

% of_nat_less_two_power
tff(fact_2321_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => ( ( N != zero_zero(nat) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),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),N)))) ) ) ) ).

% inverse_of_nat_le
tff(fact_2322_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C2))
       => ( ! [M5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M5)),X)),C2)) )
         => ( X = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2323_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ~ ! [N2: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% neg_int_cases
tff(fact_2324_zdiff__int__split,axiom,
    ! [P: fun(int,bool),X: nat,Y: nat] :
      ( pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Y))))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y)))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
         => pp(aa(int,bool,P,zero_zero(int))) ) ) ) ).

% zdiff_int_split
tff(fact_2325_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ) ).

% XOR_upper
tff(fact_2326_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_ct(nat,fun(nat,A))),divmod_nat(N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ) ).

% of_nat_code_if
tff(fact_2327_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K4: real,N: nat] :
          ( ( H != zero_zero(A) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K4))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),real_V7770717601297561774m_norm(A,H)))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2328_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,N: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cv(A,fun(A,fun(nat,fun(nat,A))),H),Z),N)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_2329_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_cw(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_2330_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,Xa2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% and_int.elims
tff(fact_2331_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))) ) )
      & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.simps
tff(fact_2332_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_ord_lessThan(A,K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),K)) ) ) ).

% lessThan_iff
tff(fact_2333_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_2334_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),X) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_2335_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_2336_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_2337_finite__lessThan,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_ord_lessThan(nat,K))) ).

% finite_lessThan
tff(fact_2338_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_lessThan(A,X)),set_ord_lessThan(A,Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% lessThan_subset_iff
tff(fact_2339_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% and_negative_int_iff
tff(fact_2340_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_2341_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),bot_bot(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_2342_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_2343_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_2344_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] : set_ord_lessThan(A,X) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_2345_infinite__Iio,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A2: A] : ~ pp(aa(set(A),bool,finite_finite2(A),set_ord_lessThan(A,A2))) ) ).

% infinite_Iio
tff(fact_2346_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_cx(A,fun(A,bool),U)) ) ).

% lessThan_def
tff(fact_2347_Iio__eq__empty__iff,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & order_bot(A) )
     => ! [N: A] :
          ( ( set_ord_lessThan(A,N) = bot_bot(set(A)) )
        <=> ( N = bot_bot(A) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_2348_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),set_ord_lessThan(A,M)),set_ord_lessThan(A,N)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N)) ) ) ).

% lessThan_strict_subset_iff
tff(fact_2349_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan(nat,N) = bot_bot(set(nat)) )
    <=> ( N = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_2350_finite__nat__bounded,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => ? [K2: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S2),set_ord_lessThan(nat,K2))) ) ).

% finite_nat_bounded
tff(fact_2351_finite__nat__iff__bounded,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S2))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S2),set_ord_lessThan(nat,K3))) ) ).

% finite_nat_iff_bounded
tff(fact_2352_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_2353_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_2354_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K)) ) ).

% and_less_eq
tff(fact_2355_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),N: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N),I) = semiri8178284476397505188at_aux(A,Inc,N,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_2356_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_2357_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
          ( ! [X4: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q,X4)),aa(A,nat,P,X4)))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),set_ord_lessThan(A,N))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),set_ord_lessThan(A,N))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_cy(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,N)) ) ) ) ).

% sum_diff_distrib
tff(fact_2358_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K5))
              & ! [N3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N3))),K5)) )
        <=> ? [N7: nat] :
            ! [N3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N3))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% lemma_NBseq_def
tff(fact_2359_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K5))
              & ! [N3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N3))),K5)) )
        <=> ? [N7: nat] :
            ! [N3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N3))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% lemma_NBseq_def2
tff(fact_2360_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_2361_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_cz(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M)) ) ).

% sum_lessThan_telescope'
tff(fact_2362_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_cl(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,M)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_2363_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_2364_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,F2: fun(nat,A),K4: A,K: nat] :
          ( ! [P4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P4),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,P4)),K4)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),K4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),K4))) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_2365_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real)))
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2366_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,X)))
        <=> ( X != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2367_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( real_V7770717601297561774m_norm(A,X) = zero_zero(real) )
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2368_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2369_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W2),N) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W2) = one_one(real) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2370_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z: A,E22: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y))),E1))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z))),E22))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2371_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_2372_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_2373_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_2374_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_2375_Suc__0__and__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),N) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Suc_0_and_eq
tff(fact_2376_and__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),N),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% and_Suc_0_eq
tff(fact_2377_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero(nat) )
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = zero_zero(nat) ) )
      & ( ~ ( ( M = zero_zero(nat) )
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_nat_unfold
tff(fact_2378_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,divide_divide(A,A2,B2)) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2379_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real))) ) ).

% norm_not_less_zero
tff(fact_2380_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,N: nat,Z: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W2),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( real_V7770717601297561774m_norm(A,W2) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2381_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,R2: real,Y: A,S: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),R2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),S))) ) ) ) ).

% norm_mult_less
tff(fact_2382_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E3))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),E3)) ) ) ).

% norm_triangle_lt
tff(fact_2383_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,R2: real,Y: A,S: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),R2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S))) ) ) ) ).

% norm_add_less
tff(fact_2384_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))
       => ~ ( ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                  & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2))))) ) )
              & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,Xa2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_2385_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_2386_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real)))
         => sums(A,aTP_Lamp_da(A,fun(nat,A),Z),divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_2387_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N)))),aa(nat,real,semiring_1_of_nat(real),binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N),N)))) ) ).

% central_binomial_lower_bound
tff(fact_2388_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A2: fun(nat,A),X: A,Y: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),X)),set_ord_atMost(nat,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),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_dd(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A2),X),Y)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff
tff(fact_2389_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)) ) )
      & ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
            | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N))) ) )
          & ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_2390_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_ord_atMost(A,K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),K)) ) ) ).

% atMost_iff
tff(fact_2391_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_2392_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_2393_finite__atMost,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_ord_atMost(nat,K))) ).

% finite_atMost
tff(fact_2394_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% sgn_greater
tff(fact_2395_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,sgn_sgn(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% sgn_less
tff(fact_2396_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,X)),set_ord_atMost(A,Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% atMost_subset_iff
tff(fact_2397_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( aa(A,A,sgn_sgn(A),A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_2398_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atMost(A,H2)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),H),H2)) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_2399_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A2),zero_zero(A)))) ) ).

% sgn_mult_self_eq
tff(fact_2400_atMost__0,axiom,
    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_2401_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_2402_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% sgn_of_nat
tff(fact_2403_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),X: A] :
          ( sums(A,aTP_Lamp_de(fun(nat,A),fun(nat,A),A2),X)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = X ) ) ) ).

% powser_sums_zero_iff
tff(fact_2404_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_2405_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_2406_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) )
        <=> ( X = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_2407_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H: A] : bot_bot(set(A)) != set_ord_atMost(A,H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_2408_infinite__Iic,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A2: A] : ~ pp(aa(set(A),bool,finite_finite2(A),set_ord_atMost(A,A2))) ) ).

% infinite_Iic
tff(fact_2409_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_df(A,fun(A,bool),U)) ) ).

% atMost_def
tff(fact_2410_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) != aa(A,A,sgn_sgn(A),A2) )
         => ( ( aa(A,A,sgn_sgn(A),A2) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_2411_atMost__atLeast0,axiom,
    ! [N: nat] : set_ord_atMost(nat,N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).

% atMost_atLeast0
tff(fact_2412_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,H)),set_or1337092689740270186AtMost(A,L2,H2))) ) ).

% not_Iic_le_Icc
tff(fact_2413_finite__nat__iff__bounded__le,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S2))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S2),set_ord_atMost(nat,K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_2414_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% sgn_1_pos
tff(fact_2415_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,A2)),set_ord_lessThan(A,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% Iic_subset_Iio_iff
tff(fact_2416_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : sums(A,aTP_Lamp_de(fun(nat,A),fun(nat,A),A2),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_2417_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% sgn_1_neg
tff(fact_2418_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = one_one(A) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).

% sgn_if
tff(fact_2419_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero(int) )
       => ( aa(int,int,sgn_sgn(int),I) = zero_zero(int) ) )
      & ( ( I != zero_zero(int) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I))
           => ( aa(int,int,sgn_sgn(int),I) = one_one(int) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I))
           => ( aa(int,int,sgn_sgn(int),I) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ) ).

% zsgn_def
tff(fact_2420_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),X)) = zero_zero(real) ) )
          & ( ( X != zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),X)) = one_one(real) ) ) ) ) ).

% norm_sgn
tff(fact_2421_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).

% sum.atMost_Suc_shift
tff(fact_2422_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_cz(fun(nat,A),fun(nat,A),F2)),set_ord_atMost(nat,I)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_2423_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,D2: fun(nat,A)] :
          ( ! [X3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),C2),X3)),set_ord_atMost(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),D2),X3)),set_ord_atMost(nat,N))
        <=> ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),N))
             => ( aa(nat,A,C2,I4) = aa(nat,A,D2,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_2424_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N),K)),binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N),N))) ).

% binomial_maximum'
tff(fact_2425_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),K6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K6)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),binomial(N,K)),binomial(N,K6))) ) ) ).

% binomial_mono
tff(fact_2426_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),K6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),K))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K6),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),binomial(N,K6)),binomial(N,K))) ) ) ) ).

% binomial_antimono
tff(fact_2427_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),binomial(N,K)),binomial(N,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% binomial_maximum
tff(fact_2428_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),N: 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_dh(fun(nat,A),fun(A,fun(nat,A)),C2),W)),set_ord_atMost(nat,N)) = zero_zero(A)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_2429_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( ! [X3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),C2),X3)),set_ord_atMost(nat,N)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),N))
             => ( aa(nat,A,C2,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_2430_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.atMost_shift
tff(fact_2431_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atMost(nat,N)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_2432_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),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),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_di(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dk(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% sum.triangle_reindex_eq
tff(fact_2433_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),binomial(N,K)),binomial(N,aa(nat,nat,suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_2434_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),K6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K6),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),binomial(N,K6)),binomial(N,K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_2435_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),K6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K6)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),binomial(N,K)),binomial(N,K6))) ) ) ).

% binomial_strict_mono
tff(fact_2436_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_dl(fun(nat,A),fun(nat,fun(A,bool)),C2),N))))
        <=> ? [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),N))
              & ( aa(nat,A,C2,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_2437_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_dl(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))) ) ) ) ).

% polyfun_roots_finite
tff(fact_2438_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: A,N: nat] :
          ( ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,N)) = zero_zero(A) )
         => ~ ! [B4: fun(nat,A)] :
                ~ ! [Z4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z4),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),set_ord_lessThan(nat,N))) ) ) ).

% polyfun_linear_factor_root
tff(fact_2439_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))) ) ) ) ).

% sum_power_shift
tff(fact_2440_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),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),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_dm(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dk(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.triangle_reindex
tff(fact_2441_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: fun(nat,A),N: nat,B2: fun(nat,A),X: A] :
          ( ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),I2))
             => ( aa(nat,A,A2,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J2))
               => ( aa(nat,A,B2,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),X)),set_ord_atMost(nat,M))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),B2),X)),set_ord_atMost(nat,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_do(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ) ).

% polynomial_product
tff(fact_2442_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,K: A] :
          ( ! [X3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),C2),X3)),set_ord_atMost(nat,N)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X3: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),set_or1337092689740270186AtMost(nat,one_one(nat),N)))
               => ( aa(nat,A,C2,X3) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_2443_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun(nat,nat),N: nat,B2: fun(nat,nat),X: nat] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),I2))
         => ( aa(nat,nat,A2,I2) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J2))
           => ( aa(nat,nat,B2,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_dp(fun(nat,nat),fun(nat,fun(nat,nat)),A2),X)),set_ord_atMost(nat,M))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_dp(fun(nat,nat),fun(nat,fun(nat,nat)),B2),X)),set_ord_atMost(nat,N))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_dr(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_2444_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K2: int,L3: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L3))
           => ( ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L3),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => pp(aa(int,bool,aa(int,fun(int,bool),P,divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L3,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,K2),L3)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_2445_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ds(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_dt(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2446_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,Z: A,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N) = A2 )
          <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_du(nat,fun(A,fun(A,fun(nat,A))),N),Z),A2)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_2447_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A2: fun(nat,A),X: A,Y: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),X)),set_ord_atMost(nat,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),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_dw(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A2),X),Y)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff_alt
tff(fact_2448_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(int) ) )
      & ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
            | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
           => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,M,N)) ) )
          & ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
           => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,M,N)),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_2449_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E3: real,C2: fun(nat,A),N: nat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
         => ? [M8: real] :
            ! [Z4: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),M8),real_V7770717601297561774m_norm(A,Z4)))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dx(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),E3),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z4)),aa(nat,nat,suc,N))))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_2450_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dy(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_2451_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dz(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_2452_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ea(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_2453_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( ( N != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eb(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_2454_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),binomial(N,K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).

% zero_less_binomial_iff
tff(fact_2455_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => sums(A,aa(A,fun(nat,A),power_power(A),C2),divide_divide(A,one_one(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_2456_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_ec(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_2457_binomial__1,axiom,
    ! [N: nat] : binomial(N,aa(nat,nat,suc,zero_zero(nat))) = N ).

% binomial_1
tff(fact_2458_binomial__0__Suc,axiom,
    ! [K: nat] : binomial(zero_zero(nat),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2459_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial(N,K) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ).

% binomial_eq_0_iff
tff(fact_2460_binomial__n__0,axiom,
    ! [N: nat] : binomial(N,zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2461_sgn__real__def,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero(real) )
       => ( aa(real,real,sgn_sgn(real),A2) = zero_zero(real) ) )
      & ( ( A2 != zero_zero(real) )
       => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
           => ( aa(real,real,sgn_sgn(real),A2) = one_one(real) ) )
          & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
           => ( aa(real,real,sgn_sgn(real),A2) = aa(real,real,uminus_uminus(real),one_one(real)) ) ) ) ) ) ).

% sgn_real_def
tff(fact_2462_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A),S: A,T2: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,G,N2)))
         => ( sums(A,F2,S)
           => ( sums(A,G,T2)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),S),T2)) ) ) ) ) ).

% sums_le
tff(fact_2463_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_2464_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_ed(nat,fun(fun(nat,A),fun(nat,A)),I),F2),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_2465_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
     => ( binomial(N,K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2466_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => ( binomial(N,K) = binomial(N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ).

% binomial_symmetric
tff(fact_2467_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),binomial(N,R2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R2))) ) ).

% binomial_le_pow
tff(fact_2468_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ee(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))
          <=> sums(A,F2,D2) ) ) ) ).

% sums_mult_iff
tff(fact_2469_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ef(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),C2))
          <=> sums(A,F2,D2) ) ) ) ).

% sums_mult2_iff
tff(fact_2470_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),binomial(N,K))) ) ).

% zero_less_binomial
tff(fact_2471_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),binomial(N,M)),binomial(M,K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),binomial(N,K)),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% choose_mult
tff(fact_2472_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A2: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eg(A,fun(fun(nat,A),fun(nat,A)),C2),F2),A2)
         => ( ( C2 != zero_zero(A) )
           => sums(A,F2,divide_divide(A,A2,C2)) ) ) ) ).

% sums_mult_D
tff(fact_2473_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_eh(fun(nat,A),fun(nat,A),F2),S)
           => sums(A,F2,S) ) ) ) ).

% sums_Suc_imp
tff(fact_2474_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_ei(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_2475_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,aTP_Lamp_eh(fun(nat,A),fun(nat,A),F2),S)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_2476_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [N: nat,F2: fun(nat,A),S: A] :
          ( ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
             => ( aa(nat,A,F2,I2) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ej(nat,fun(fun(nat,A),fun(nat,A)),N),F2),S)
          <=> sums(A,F2,S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_2477_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N4: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),N4))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),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_2478_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,bool),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),P)))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ek(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),P),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(fun(nat,bool),set(nat),collect(nat),P))) ) ) ).

% sums_If_finite
tff(fact_2479_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_el(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A3)) ) ) ).

% sums_If_finite_set
tff(fact_2480_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_em(nat,fun(A,fun(nat,A)),M),Z),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)) ) ).

% powser_sums_if
tff(fact_2481_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S2: A,A3: set(nat),S5: A,F2: fun(nat,A)] :
          ( sums(A,G,S2)
         => ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
           => ( ( S5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),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_en(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A3)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_eo(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F2),S5) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_2482_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),N),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),binomial(N,K)))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2483_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),binomial(N,K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% binomial_le_pow2
tff(fact_2484_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( binomial(N,K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2485_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),binomial(N,K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2486_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_ep(nat,fun(nat,fun(nat,nat)),M),N)),set_ord_atMost(nat,M)) = binomial(aa(nat,nat,suc,N),M) ) ) ).

% sum_choose_diagonal
tff(fact_2487_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( binomial(N,aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(nat,nat,suc,K))),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_2488_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
               => pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2)) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,I2),J2)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_2489_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_eq(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),aa(num,num,bit0,one2)))),gbinomial(A,R2,aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_2490_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,U,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,U,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_2491_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_le
tff(fact_2492_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X)) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2493_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A)))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2494_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% real_sqrt_less_iff
tff(fact_2495_gbinomial__0_I2_J,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [K: nat] : gbinomial(B,zero_zero(B),aa(nat,nat,suc,K)) = zero_zero(B) ) ).

% gbinomial_0(2)
tff(fact_2496_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) ) ).

% real_sqrt_gt_0_iff
tff(fact_2497_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% real_sqrt_lt_0_iff
tff(fact_2498_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_2499_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : gbinomial(A,A2,zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_2500_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : gbinomial(A,A2,aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% gbinomial_Suc0
tff(fact_2501_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),Y)) ) ).

% real_sqrt_gt_1_iff
tff(fact_2502_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ).

% real_sqrt_lt_1_iff
tff(fact_2503_log__eq__one,axiom,
    ! [A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_2504_log__less__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)))
          <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_2505_log__less__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(A2),X)),one_one(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),A2)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_2506_one__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_2507_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(A2),X)),zero_zero(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_2508_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_2509_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_le_zero
tff(fact_2510_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% zero_less_ceiling
tff(fact_2511_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% ceiling_le_numeral
tff(fact_2512_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_less_one
tff(fact_2513_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),V2)),X)) ) ) ).

% numeral_less_ceiling
tff(fact_2514_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% one_le_ceiling
tff(fact_2515_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ).

% ceiling_le_one
tff(fact_2516_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ).

% one_less_ceiling
tff(fact_2517_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_2518_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(A2),X)),zero_zero(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_2519_one__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_2520_log__le__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(A2),X)),one_one(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),A2)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_2521_log__le__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)))
          <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_2522_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).

% ceiling_less_zero
tff(fact_2523_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X)) ) ) ).

% zero_le_ceiling
tff(fact_2524_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_2525_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).

% ceiling_less_numeral
tff(fact_2526_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).

% numeral_le_ceiling
tff(fact_2527_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2528_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X)) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2529_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))) ) ).

% real_sqrt_less_mono
tff(fact_2530_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_2531_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,X))) ) ) ).

% ceiling_mono
tff(fact_2532_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ).

% le_of_int_ceiling
tff(fact_2533_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% ceiling_less_cancel
tff(fact_2534_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% ceiling_le_iff
tff(fact_2535_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A2)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),A2)) ) ) ).

% ceiling_le
tff(fact_2536_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),X)) ) ) ).

% less_ceiling_iff
tff(fact_2537_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N),K) = gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_2538_sqrt2__less__2,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% sqrt2_less_2
tff(fact_2539_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A)))) ) ).

% of_int_ceiling_le_add_one
tff(fact_2540_log__base__change,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(B2),X) = divide_divide(real,aa(real,real,log(A2),X),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_2541_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),one_one(A))),R2)) ) ).

% of_int_ceiling_diff_one_le
tff(fact_2542_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_2543_less__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)),M))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),M))) ) ) ).

% less_log_of_power
tff(fact_2544_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),K))),K)),gbinomial(A,A2,K))) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_2545_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_2546_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P,archimedean_ceiling(A,T2)))
        <=> ! [I4: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),aa(int,A,ring_1_of_int(A),I4))) )
             => pp(aa(int,bool,P,I4)) ) ) ) ).

% ceiling_split
tff(fact_2547_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archimedean_ceiling(A,X) = A2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A2))) ) ) ) ).

% ceiling_eq_iff
tff(fact_2548_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z)))
           => ( archimedean_ceiling(A,X) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_2549_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))),one_one(A))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ) ).

% ceiling_correct
tff(fact_2550_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le
tff(fact_2551_log__mult,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_2552_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))) ) ) ).

% ceiling_less_iff
tff(fact_2553_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X)) ) ) ).

% le_ceiling_iff
tff(fact_2554_log__divide,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
           => ( aa(real,real,log(A2),divide_divide(real,X,Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_2555_le__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)),M))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),M))) ) ) ).

% le_log_of_power
tff(fact_2556_log__base__pow,axiom,
    ! [A2: real,N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( aa(real,real,log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),N)),X) = divide_divide(real,aa(real,real,log(A2),X),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ).

% log_base_pow
tff(fact_2557_log__nat__power,axiom,
    ! [X: real,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log(B2),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),X)) ) ) ).

% log_nat_power
tff(fact_2558_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A2,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,A2,K)),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_2559_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_2560_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_2561_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P2,Q4)))),Q4))) ) ) ).

% ceiling_divide_upper
tff(fact_2562_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),U))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_2563_log__of__power__less,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_less
tff(fact_2564_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) )
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_2565_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_2566_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P2,Q4)))),one_one(A))),Q4)),P2)) ) ) ).

% ceiling_divide_lower
tff(fact_2567_log__of__power__le,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_le
tff(fact_2568_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),N)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),N)),one_one(A))))
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_2569_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
         => ( gbinomial(A,A2,K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_2570_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))))) ).

% arsinh_real_aux
tff(fact_2571_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),M))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M)))) ) ).

% less_log2_of_power
tff(fact_2572_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),M))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M)))) ) ).

% le_log2_of_power
tff(fact_2573_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_er(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_2574_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_less
tff(fact_2575_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
         => ( gbinomial(A,A2,K) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),K))),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_2576_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) )
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_2577_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),X))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat)))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_2578_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_2579_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_2580_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),Y)),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_2581_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_2582_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_idempotent
tff(fact_2583_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_2584_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_2585_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2586_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_2587_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_2588_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus_cancel
tff(fact_2589_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% abs_of_nat
tff(fact_2590_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_2591_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_2592_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),zero_zero(A)))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2593_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% abs_le_self_iff
tff(fact_2594_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_2595_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A2)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2596_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = one_one(A) ) ) ) ) ).

% powr_zero_eq_one
tff(fact_2597_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_2598_powr__gt__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),powr(real,X,A2)))
    <=> ( X != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_2599_powr__less__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,X,A2)),powr(real,X,B2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2)) ) ) ).

% powr_less_cancel_iff
tff(fact_2600_sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F2: fun(A,B),A3: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_es(fun(A,B),fun(A,B),F2)),A3))) ) ).

% sum_abs
tff(fact_2601_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,aa(A,A,abs_abs(A),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2602_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A2,aa(A,A,abs_abs(A),B2))),zero_zero(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2603_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_nonpos
tff(fact_2604_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_2605_powr__eq__one__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( ( powr(real,A2,X) = one_one(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_2606_powr__le__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,X,A2)),powr(real,X,B2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2)) ) ) ).

% powr_le_cancel_iff
tff(fact_2607_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),X)) ) ) ).

% artanh_minus_real
tff(fact_2608_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A2),zero_zero(A)))) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_2609_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A2),zero_zero(A)))) ) ).

% sgn_abs
tff(fact_2610_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F2: fun(A,B),A3: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_es(fun(A,B),fun(A,B),F2)),A3))) ) ).

% sum_abs_ge_zero
tff(fact_2611_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),N)))
        <=> ( ( A2 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2612_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ) ).

% zero_le_floor
tff(fact_2613_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A))) ) ) ).

% floor_less_zero
tff(fact_2614_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),V2)),X)) ) ) ).

% numeral_le_floor
tff(fact_2615_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% zero_less_floor
tff(fact_2616_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_le_zero
tff(fact_2617_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% floor_less_numeral
tff(fact_2618_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% one_le_floor
tff(fact_2619_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_less_one
tff(fact_2620_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),powr(real,A2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_2621_powr__log__cancel,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( powr(real,A2,aa(real,real,log(A2),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_2622_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).

% numeral_less_floor
tff(fact_2623_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).

% floor_le_numeral
tff(fact_2624_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) ) ) ).

% one_less_floor
tff(fact_2625_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ).

% floor_le_one
tff(fact_2626_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X)) ) ) ).

% neg_numeral_le_floor
tff(fact_2627_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)))) ) ) ).

% floor_less_neg_numeral
tff(fact_2628_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X)) ) ) ).

% neg_numeral_less_floor
tff(fact_2629_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A)))) ) ) ).

% floor_le_neg_numeral
tff(fact_2630_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% abs_le_D1
tff(fact_2631_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_self
tff(fact_2632_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_2633_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ).

% abs_minus_commute
tff(fact_2634_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_zero
tff(fact_2635_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))) ) ) ).

% floor_mono
tff(fact_2636_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_2637_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A2)),zero_zero(A))) ) ).

% abs_not_less_zero
tff(fact_2638_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)) ) ).

% of_int_floor_le
tff(fact_2639_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% floor_less_cancel
tff(fact_2640_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq
tff(fact_2641_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A2)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),B2)),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))) ) ) ) ).

% abs_mult_less
tff(fact_2642_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2643_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)))) ) ).

% abs_triangle_ineq3
tff(fact_2644_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)))) ) ).

% abs_triangle_ineq2
tff(fact_2645_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,abs_abs(A),A2),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_2646_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)) ) ) ) ).

% abs_leI
tff(fact_2647_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ).

% abs_le_D2
tff(fact_2648_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ) ).

% abs_le_iff
tff(fact_2649_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_minus_self
tff(fact_2650_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A2)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ) ).

% abs_less_iff
tff(fact_2651_powr__less__mono2__neg,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,Y,A2)),powr(real,X,A2))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_2652_powr__non__neg,axiom,
    ! [A2: real,X: real] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,A2,X)),zero_zero(real))) ).

% powr_non_neg
tff(fact_2653_powr__less__mono,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,X,A2)),powr(real,X,B2))) ) ) ).

% powr_less_mono
tff(fact_2654_powr__less__cancel,axiom,
    ! [X: real,A2: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,X,A2)),powr(real,X,B2)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2)) ) ) ).

% powr_less_cancel
tff(fact_2655_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E2)) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2656_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_2657_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_2658_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A2))),zero_zero(A))) ) ).

% abs_minus_le_zero
tff(fact_2659_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2660_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2661_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( divide_divide(A,aa(A,A,abs_abs(A),X),Y) = aa(A,A,abs_abs(A),divide_divide(A,X,Y)) ) ) ) ).

% abs_div_pos
tff(fact_2662_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),N))) ) ).

% zero_le_power_abs
tff(fact_2663_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X2) = aa(A,A,uminus_uminus(A),X2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X2) = X2 ) ) ) ) ).

% abs_if_raw
tff(fact_2664_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_neg
tff(fact_2665_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ) ).

% abs_if
tff(fact_2666_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2))))) ) ).

% abs_diff_triangle_ineq
tff(fact_2667_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq4
tff(fact_2668_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),R2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2))) ) ) ) ).

% abs_diff_le_iff
tff(fact_2669_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),R2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2))) ) ) ) ).

% abs_diff_less_iff
tff(fact_2670_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),X)) ) ) ).

% le_floor_iff
tff(fact_2671_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% floor_less_iff
tff(fact_2672_powr__mono2_H,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,Y,A2)),powr(real,X,A2))) ) ) ) ).

% powr_mono2'
tff(fact_2673_powr__less__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,X,A2)),powr(real,Y,A2))) ) ) ) ).

% powr_less_mono2
tff(fact_2674_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),powr(real,X,Y))) ) ) ).

% gr_one_powr
tff(fact_2675_powr__inj,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,X) = powr(real,A2,Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_2676_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( ( A2 = zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = zero_zero(A) ) )
          & ( ( A2 != zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = one_one(A) ) ) ) ) ).

% abs_sgn_eq
tff(fact_2677_abs__real__def,axiom,
    ! [A2: real] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A2) = aa(real,real,uminus_uminus(real),A2) ) )
      & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A2) = A2 ) ) ) ).

% abs_real_def
tff(fact_2678_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X)))) ) ).

% abs_add_one_gt_zero
tff(fact_2679_floor__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),X)) = K )
        <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,B2,aa(int,real,ring_1_of_int(real),K))),X))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),powr(real,B2,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_2680_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% of_int_leD
tff(fact_2681_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% of_int_lessD
tff(fact_2682_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( powr(real,X,aa(nat,real,semiring_1_of_nat(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N) ) ) ).

% powr_realpow
tff(fact_2683_less__log__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(real,real,log(B2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,B2,Y)),X)) ) ) ) ).

% less_log_iff
tff(fact_2684_log__less__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(B2),X)),Y))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),powr(real,B2,Y))) ) ) ) ).

% log_less_iff
tff(fact_2685_less__powr__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),powr(real,B2,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(B2),X)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_2686_powr__less__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,B2,Y)),X))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(real,real,log(B2),X))) ) ) ) ).

% powr_less_iff
tff(fact_2687_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(int,real,ring_1_of_int(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))))
       => ( archim6421214686448440834_floor(real,X) = N ) ) ) ).

% floor_eq
tff(fact_2688_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real)))) ).

% real_of_int_floor_add_one_gt
tff(fact_2689_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_2690_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z))))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),aa(int,A,ring_1_of_int(A),M))))) ) ).

% round_diff_minimal
tff(fact_2691_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))))
           => ( archim6421214686448440834_floor(A,X) = Z ) ) ) ) ).

% floor_unique
tff(fact_2692_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,X) = A2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A)))) ) ) ) ).

% floor_eq_iff
tff(fact_2693_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T2)))
        <=> ! [I4: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A)))) )
             => pp(aa(int,bool,P,I4)) ) ) ) ).

% floor_split
tff(fact_2694_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor
tff(fact_2695_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X)) ) ) ).

% less_floor_iff
tff(fact_2696_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))) ) ) ).

% floor_le_iff
tff(fact_2697_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))))) ) ) ).

% floor_correct
tff(fact_2698_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_le_square_iff
tff(fact_2699_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,uminus_uminus(real),one_one(real))) = divide_divide(real,one_one(real),X) ) ) ).

% powr_neg_one
tff(fact_2700_powr__le__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,B2,Y)),X))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,log(B2),X))) ) ) ) ).

% powr_le_iff
tff(fact_2701_le__powr__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),powr(real,B2,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(B2),X)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_2702_log__le__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(B2),X)),Y))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),powr(real,B2,Y))) ) ) ) ).

% log_le_iff
tff(fact_2703_le__log__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,log(B2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,B2,Y)),X)) ) ) ) ).

% le_log_iff
tff(fact_2704_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(int,real,ring_1_of_int(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))))
       => ( archim6421214686448440834_floor(real,X) = N ) ) ) ).

% floor_eq2
tff(fact_2705_sgn__power__injE,axiom,
    ! [A2: real,N: nat,X: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A2)),N)) = X )
     => ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),N)) )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_2706_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P2,Q4)))),Q4)),P2)) ) ) ).

% floor_divide_lower
tff(fact_2707_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y)) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2708_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,bool)),X: A] :
          ( ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X4))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,X4),aa(nat,A,aa(A,fun(nat,A),power_power(A),X4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(A,A,abs_abs(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_sqrt_wlog
tff(fact_2709_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_le_1
tff(fact_2710_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_less_1
tff(fact_2711_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono_even
tff(fact_2712_add__log__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(B2),X)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_2713_log__add__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B2),X)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_2714_minus__log__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),aa(real,real,log(B2),X)) = aa(real,real,log(B2),divide_divide(real,powr(real,B2,Y),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_2715_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),X: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I2))) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I6) = one_one(B) )
           => ( ! [I2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,I2)),B2))),Delta)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_et(fun(A,B),fun(fun(A,B),fun(A,B)),X),A2)),I6)),B2))),Delta)) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2716_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q4: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P2,Q4)))),one_one(A))),Q4))) ) ) ).

% floor_divide_upper
tff(fact_2717_log__minus__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(B2),X)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_2718_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),N))) = divide_divide(real,one_one(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),N))) ) ) ).

% powr_neg_numeral
tff(fact_2719_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),X))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% of_int_round_abs_le
tff(fact_2720_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),N)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
         => ( archimedean_round(A,X) = N ) ) ) ).

% round_unique'
tff(fact_2721_lemma__interval,axiom,
    ! [A2: real,X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [Y3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y3))),D3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),B2)) ) ) ) ) ) ).

% lemma_interval
tff(fact_2722_lemma__interval__lt,axiom,
    ! [A2: real,X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [Y3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y3))),D3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),B2)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2723_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archimedean_ceiling(A,X) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).

% round_altdef
tff(fact_2724_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),arctan(X)) = arctan(divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% arctan_double
tff(fact_2725_powr__int,axiom,
    ! [X: real,I: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
         => ( powr(real,X,aa(int,real,ring_1_of_int(real),I)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(int,nat,nat2,I)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
         => ( powr(real,X,aa(int,real,ring_1_of_int(real),I)) = divide_divide(real,one_one(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I)))) ) ) ) ) ).

% powr_int
tff(fact_2726_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list(bool)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(list(bool),int,aa(int,fun(list(bool),int),aa(fun(bool,int),fun(int,fun(list(bool),int)),groups4207007520872428315er_sum(bool,int),zero_neq_one_of_bool(int)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(list(bool),nat,size_size(list(bool)),Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_2727_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int)))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_2728_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),arctan(X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% arctan_less_zero_iff
tff(fact_2729_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),arctan(X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% zero_less_arctan_iff
tff(fact_2730_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_2731_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_2732_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa(int,nat,nat2,I) = zero_zero(nat) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int))) ) ).

% nat_0_iff
tff(fact_2733_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
     => ( aa(int,nat,nat2,Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_2734_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z)) ) ) ).

% zless_nat_conj
tff(fact_2735_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_2736_nat__zminus__int,axiom,
    ! [N: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_2737_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

% zero_less_nat_eq
tff(fact_2738_nat__ceiling__le__eq,axiom,
    ! [X: real,A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,X))),A2))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),A2))) ) ).

% nat_ceiling_le_eq
tff(fact_2739_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z)) ) ).

% one_less_nat_eq
tff(fact_2740_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2741_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2742_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2743_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_2744_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),arctan(X)),arctan(Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% arctan_less_iff
tff(fact_2745_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),arctan(X)),arctan(Y))) ) ).

% arctan_monotone
tff(fact_2746_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))) ).

% nat_abs_triangle_ineq
tff(fact_2747_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
       => ( aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A2) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
       => ( aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_2748_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_2749_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Y))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y))) ) ).

% nat_mono
tff(fact_2750_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite2(int),S2))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M3),aa(int,int,abs_abs(int),N3)))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),N3),S2)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_2751_infinite__int__iff__unbounded,axiom,
    ! [S2: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite2(int),S2))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M3),aa(int,int,abs_abs(int),N3)))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),N3),S2)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_2752_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,X))) ) ).

% frac_ge_0
tff(fact_2753_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),archimedean_frac(A,X)),one_one(A))) ) ).

% frac_lt_1
tff(fact_2754_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z)) ) ) ).

% nat_mono_iff
tff(fact_2755_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R2),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R2))))) ) ).

% of_nat_ceiling
tff(fact_2756_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(int,nat,nat2,Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_2757_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),N))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_le_iff
tff(fact_2758_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I) = aa(int,int,uminus_uminus(int),I) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I) = I ) ) ) ).

% zabs_def
tff(fact_2759_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),modulo_modulo(int,K,L))),aa(int,int,abs_abs(int),L))) ) ).

% abs_mod_less
tff(fact_2760_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R2)))),R2)) ) ) ).

% of_nat_floor
tff(fact_2761_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_2762_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),W2))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_2763_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa(int,nat,nat2,W2) )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( W2 = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff2
tff(fact_2764_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa(int,nat,nat2,W2) = M )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( W2 = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff
tff(fact_2765_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,archim6421214686448440834_floor(A,A2))),aa(int,nat,nat2,archim6421214686448440834_floor(A,B2)))),aa(int,nat,nat2,archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))))) ) ).

% le_mult_nat_floor
tff(fact_2766_split__nat,axiom,
    ! [P: fun(nat,bool),I: int] :
      ( pp(aa(nat,bool,P,aa(int,nat,nat2,I)))
    <=> ( ! [N3: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N3) )
           => pp(aa(nat,bool,P,N3)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
         => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ) ).

% split_nat
tff(fact_2767_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(int,nat,nat2,K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ) ) ).

% le_nat_iff
tff(fact_2768_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_2769_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_2770_le__nat__floor,axiom,
    ! [X: nat,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),X)),A2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(int,nat,nat2,archim6421214686448440834_floor(real,A2)))) ) ).

% le_nat_floor
tff(fact_2771_nat__2,axiom,
    aa(int,nat,nat2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_2772_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),M))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(nat,int,semiring_1_of_nat(int),M))) ) ) ).

% nat_less_iff
tff(fact_2773_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% frac_eq
tff(fact_2774_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_2775_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)) ) ) ) ) ).

% frac_add
tff(fact_2776_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z6),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z6)) = aa(int,nat,nat2,Z) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z6),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z6)) = if(nat,aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z6)),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z6))) ) ) ) ).

% diff_nat_eq_if
tff(fact_2777_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K) = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K) = aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K)) ) ) ) ) ).

% of_int_of_nat
tff(fact_2778_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,Z)),M))
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z),aa(nat,int,semiring_1_of_nat(int),M))) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_dvd_iff
tff(fact_2779_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N)) )
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,M)),K))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F2,N)))
           => ? [I2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I2))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
                & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2780_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Z))),one_one(int))),D2)))) ) ).

% incr_lemma
tff(fact_2781_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Z))),one_one(int))),D2))),Z)) ) ).

% decr_lemma
tff(fact_2782_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q4: int] :
      ( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),L) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),L)))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_2783_nat__ivt__aux,axiom,
    ! [N: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F2,N)))
         => ? [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2784_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3) )
               => ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),A22) )
                 => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),A22)))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_2785_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K3) ) )
        | ? [L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L4) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4)))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_2786_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int)) ) ) ) ) ).

% floor_add
tff(fact_2787_nat0__intermed__int__val,axiom,
    ! [N: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F2,I2)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F2,N)))
         => ? [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2788_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),arctan(X)),arctan(Y)) = arctan(divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)))) ) ) ) ).

% arctan_add
tff(fact_2789_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
         => ( powr(real,X,aa(int,real,ring_1_of_int(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(int,nat,nat2,N)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
         => ( powr(real,X,aa(int,real,ring_1_of_int(real),N)) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_2790_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_2791_the__elem__eq,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ).

% the_elem_eq
tff(fact_2792_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),divide_divide(real,X,aa(nat,real,semiring_1_of_nat(real),N)))),N)),exp(real,aa(real,real,uminus_uminus(real),X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_2793_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,X,aa(nat,real,semiring_1_of_nat(real),N)))),N)),exp(real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_2794_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I6: set(A),F2: fun(A,B),I: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),F4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_eu(set(A),fun(fun(A,B),fun(A,bool)),I6),F2))),F4))
           => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I6))
               => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,I6)),aa(A,B,F2,I)) ) )
              & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I6))
               => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F2,I6) ) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_2795_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_2796_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_2797_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),exp(real,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% exp_less_cancel_iff
tff(fact_2798_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),exp(real,Y))) ) ).

% exp_less_mono
tff(fact_2799_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_2800_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_2801_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_2802_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_2803_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% inverse_negative_iff_negative
tff(fact_2804_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% inverse_positive_iff_positive
tff(fact_2805_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_2806_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P2,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_2807_sum_Oeq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I6: set(B),P2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( groups1027152243600224163dd_sum(B,A,P2,I6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P2),I6) ) ) ) ).

% sum.eq_sum
tff(fact_2808_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_2809_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_2810_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),A2)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_2811_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_2812_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),exp(real,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% one_less_exp_iff
tff(fact_2813_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% exp_less_one_iff
tff(fact_2814_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I6: set(B),P2: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_az(set(B),fun(fun(B,A),fun(B,bool)),I6),P2))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I6))
             => ( groups1027152243600224163dd_sum(B,A,P2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I),I6)) = groups1027152243600224163dd_sum(B,A,P2,I6) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I6))
             => ( groups1027152243600224163dd_sum(B,A,P2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I),I6)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P2,I)),groups1027152243600224163dd_sum(B,A,P2,I6)) ) ) ) ) ) ).

% sum.insert'
tff(fact_2815_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A2)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A2)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_2816_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_2817_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
         => ( A2 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_2818_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( ( A2 != zero_zero(A) )
           => ( ( B2 != zero_zero(A) )
             => ( A2 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_2819_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_2820_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_2821_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),exp(real,Y)))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% exp_less_cancel
tff(fact_2822_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : exp(A,X) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_2823_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I6: set(B)] : groups1027152243600224163dd_sum(B,A,G,aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aTP_Lamp_ev(fun(B,A),fun(set(B),fun(B,bool)),G),I6))) = groups1027152243600224163dd_sum(B,A,G,I6) ) ).

% sum.non_neutral'
tff(fact_2824_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),R2),real_V7770717601297561774m_norm(A,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),X))),aa(real,real,inverse_inverse(real),R2))) ) ) ) ).

% norm_inverse_le_norm
tff(fact_2825_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% inverse_less_imp_less
tff(fact_2826_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% less_imp_inverse_less
tff(fact_2827_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_2828_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_2829_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
         => ( ( A2 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_2830_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
         => ( ( A2 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_2831_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))) ) ) ).

% negative_imp_inverse_negative
tff(fact_2832_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ).

% positive_imp_inverse_positive
tff(fact_2833_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_2834_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_2835_not__exp__less__zero,axiom,
    ! [X: real] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),zero_zero(real))) ).

% not_exp_less_zero
tff(fact_2836_exp__gt__zero,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),exp(real,X))) ).

% exp_gt_zero
tff(fact_2837_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ? [X4: real] : exp(real,X4) = Y ) ).

% exp_total
tff(fact_2838_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_2839_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% inverse_le_imp_le
tff(fact_2840_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% le_imp_inverse_le
tff(fact_2841_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_2842_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_2843_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% inverse_le_1_iff
tff(fact_2844_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% one_less_inverse
tff(fact_2845_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% one_less_inverse_iff
tff(fact_2846_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,inverse_inverse(A),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_2847_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_2848_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_2849_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_2850_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = divide_divide(A,one_one(A),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_2851_sum_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I6: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ew(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G,I6)),groups1027152243600224163dd_sum(B,A,H,I6)) ) ) ) ).

% sum.distrib_triv'
tff(fact_2852_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),exp(real,X))) ) ).

% exp_gt_one
tff(fact_2853_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
               => ( aa(B,A,G,X4) = zero_zero(A) ) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                 => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
             => ( groups1027152243600224163dd_sum(B,A,G,T3) = groups1027152243600224163dd_sum(B,A,H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_2854_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
               => ( aa(B,A,H,I2) = zero_zero(A) ) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                 => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
             => ( groups1027152243600224163dd_sum(B,A,G,S2) = groups1027152243600224163dd_sum(B,A,H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_2855_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
               => ( aa(B,A,G,X4) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G,T3) = groups1027152243600224163dd_sum(B,A,G,S2) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_2856_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
               => ( aa(B,A,G,X4) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G,S2) = groups1027152243600224163dd_sum(B,A,G,T3) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_2857_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ) ).

% inverse_le_iff
tff(fact_2858_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ) ).

% inverse_less_iff
tff(fact_2859_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% one_le_inverse
tff(fact_2860_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% inverse_less_1_iff
tff(fact_2861_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ) ).

% one_le_inverse_iff
tff(fact_2862_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_2863_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),X)) ) ) ).

% reals_Archimedean
tff(fact_2864_sum_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I6: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_az(set(B),fun(fun(B,A),fun(B,bool)),I6),G))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_az(set(B),fun(fun(B,A),fun(B,bool)),I6),H))))
           => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ew(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G,I6)),groups1027152243600224163dd_sum(B,A,H,I6)) ) ) ) ) ).

% sum.distrib'
tff(fact_2865_forall__pos__mono__1,axiom,
    ! [P: fun(real,bool),E3: real] :
      ( ! [D3: real,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D3),E2))
         => ( pp(aa(real,bool,P,D3))
           => pp(aa(real,bool,P,E2)) ) )
     => ( ! [N2: nat] : pp(aa(real,bool,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
         => pp(aa(real,bool,P,E3)) ) ) ) ).

% forall_pos_mono_1
tff(fact_2866_forall__pos__mono,axiom,
    ! [P: fun(real,bool),E3: real] :
      ( ! [D3: real,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D3),E2))
         => ( pp(aa(real,bool,P,D3))
           => pp(aa(real,bool,P,E2)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero(nat) )
           => pp(aa(real,bool,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N2)))) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
         => pp(aa(real,bool,P,E3)) ) ) ) ).

% forall_pos_mono
tff(fact_2867_real__arch__inverse,axiom,
    ! [E3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
    <=> ? [N3: nat] :
          ( ( N3 != zero_zero(nat) )
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N3))))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N3))),E3)) ) ) ).

% real_arch_inverse
tff(fact_2868_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [I6: set(B),P2: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_az(set(B),fun(fun(B,A),fun(B,bool)),I6),P2))))
           => ( groups1027152243600224163dd_sum(B,A,P2,I6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P2),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_az(set(B),fun(fun(B,A),fun(B,bool)),I6),P2))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_az(set(B),fun(fun(B,A),fun(B,bool)),I6),P2))))
           => ( groups1027152243600224163dd_sum(B,A,P2,I6) = zero_zero(A) ) ) ) ) ).

% sum.G_def
tff(fact_2869_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F2: fun(fun(A,B),C),G: C] :
          ( ! [X4: fun(A,B)] : aa(fun(A,B),C,F2,X4) = G
         => ( aa(fun(A,B),C,F2,aTP_Lamp_ex(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_2870_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N2))),X)) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_2871_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_2872_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,divide_divide(A,X,aa(nat,A,semiring_1_of_nat(A),N)))),N) = exp(A,X) ) ) ) ).

% exp_divide_power_eq
tff(fact_2873_log__inverse,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),X)) ) ) ) ) ).

% log_inverse
tff(fact_2874_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X)))) ) ).

% plus_inverse_ge_2
tff(fact_2875_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(real,real,inverse_inverse(real),X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_2876_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I6: set(A),F2: fun(A,B),I: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_eu(set(A),fun(fun(A,B),fun(A,bool)),I6),F2))))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I6))
             => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,I6)),aa(A,B,F2,I)) ) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I6))
             => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F2,I6) ) ) ) ) ) ).

% sum_diff1'
tff(fact_2877_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero(real) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ? [T7: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T7)))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),X)))
            & ( exp(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ey(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,exp(real,T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_2878_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),X) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,ln_ln(real),exp(real,one_one(real))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq1
tff(fact_2879_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A,A2: A] :
          ( ( ( X = zero_zero(A) )
           => ( powr(A,X,A2) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( powr(A,X,A2) = exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),X))) ) ) ) ) ).

% powr_def
tff(fact_2880_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),exp(real,one_one(real)))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq2
tff(fact_2881_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_2882_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2883_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_2884_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_2885_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,ln_ln(real),Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_2886_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_2887_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_2888_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_2889_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( aa(real,real,ln_ln(real),X) = zero_zero(real) )
      <=> ( X = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_2890_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_2891_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ).

% ln_less_zero_iff
tff(fact_2892_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( exp(real,aa(real,real,ln_ln(real),X)) = X )
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% exp_ln_iff
tff(fact_2893_exp__ln,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( exp(real,aa(real,real,ln_ln(real),X)) = X ) ) ).

% exp_ln
tff(fact_2894_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ).

% ln_le_zero_iff
tff(fact_2895_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_2896_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_2897_ln__less__self,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),X)) ) ).

% ln_less_self
tff(fact_2898_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_zero
tff(fact_2899_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,N)),zero_zero(A))) ) ).

% fact_not_neg
tff(fact_2900_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_gt_zero
tff(fact_2901_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_1
tff(fact_2902_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ).

% fact_mono
tff(fact_2903_cosh__real__pos,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),cosh(real,X))) ).

% cosh_real_pos
tff(fact_2904_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,M))) ) ) ).

% fact_dvd
tff(fact_2905_ln__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),X)) ) ).

% ln_bound
tff(fact_2906_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X))) ) ).

% ln_gt_zero
tff(fact_2907_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),zero_zero(real))) ) ) ).

% ln_less_zero
tff(fact_2908_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_2909_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ) ).

% fact_less_mono
tff(fact_2910_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),zero_zero(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_2911_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_2912_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cosh(real,X)),cosh(real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_2913_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( modulo_modulo(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_2914_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,N)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),N)))) ) ).

% fact_le_power
tff(fact_2915_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_2916_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( cosh(real,aa(real,real,ln_ln(real),X)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% cosh_ln_real
tff(fact_2917_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_2918_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,aa(real,fun(real,real),minus_minus(real),X),one_one(real)) )
       => ( X = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_2919_ln__div,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( aa(real,real,ln_ln(real),divide_divide(real,X,Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_2920_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,Y)),X)) ) ) ).

% ln_ge_iff
tff(fact_2921_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_inverse
tff(fact_2922_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),semiring_char_0_fact(A,N))) ) ) ).

% choose_dvd
tff(fact_2923_ln__2__less__1,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),one_one(real))) ).

% ln_2_less_1
tff(fact_2924_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),one_one(real)))) ) ).

% ln_le_minus_one
tff(fact_2925_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y))),divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y),Y))) ) ) ).

% ln_diff_le
tff(fact_2926_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_2927_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_realpow
tff(fact_2928_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),X))),aa(real,real,uminus_uminus(real),X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_2929_ln__powr__bound,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),divide_divide(real,powr(real,X,A2),A2))) ) ) ).

% ln_powr_bound
tff(fact_2930_ln__powr__bound2,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),X),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),X))) ) ) ).

% ln_powr_bound2
tff(fact_2931_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
         => ( ( B2 != one_one(real) )
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
             => ( aa(real,real,log(A2),X) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,ln_ln(real),B2),aa(real,real,ln_ln(real),A2))),aa(real,real,log(B2),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2932_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( ( M = zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ) ).

% fact_num_eq_if
tff(fact_2933_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_2934_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,semiring_1_of_nat(A),binomial(N,K)) = divide_divide(A,semiring_char_0_fact(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ) ).

% binomial_fact
tff(fact_2935_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),binomial(N,K))) = divide_divide(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).

% fact_binomial
tff(fact_2936_ln__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,X)) = divide_divide(real,aa(real,real,ln_ln(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% ln_sqrt
tff(fact_2937_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: real,N: nat,Diff: fun(nat,fun(A,real))] :
          ( ( X = zero_zero(real) )
         => ( ( N != zero_zero(nat) )
           => ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ez(real,fun(fun(nat,fun(A,real)),fun(nat,real)),X),Diff)),set_ord_lessThan(nat,N)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_2938_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ? [B8: real] : aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_fa(real,fun(fun(nat,real),fun(nat,real)),H),J)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),B8),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N),semiring_char_0_fact(real,N)))) ) ).

% Maclaurin_lemma
tff(fact_2939_sin__coeff__def,axiom,
    ! [X2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X2))
       => ( sin_coeff(X2) = zero_zero(real) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X2))
       => ( sin_coeff(X2) = divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X2),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),semiring_char_0_fact(real,X2)) ) ) ) ).

% sin_coeff_def
tff(fact_2940_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( ( ( K = zero_zero(nat) )
           => ( gbinomial(A,A2,K) = one_one(A) ) )
          & ( ( K != zero_zero(nat) )
           => ( gbinomial(A,A2,K) = divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_fb(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A)),semiring_char_0_fact(A,K)) ) ) ) ) ).

% gbinomial_code
tff(fact_2941_ln__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,real,ln_ln(real),X) = suminf(real,aTP_Lamp_fc(real,fun(nat,real),X)) ) ) ) ).

% ln_series
tff(fact_2942_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),X)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2943_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_fe(A,fun(A,fun(nat,A)),X),Y),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_2944_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,B2: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,B2),X) )
        <=> ( ( A2 = B2 )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_2945_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real] : aa(A,A,real_V8093663219630862766scaleR(A,A2),zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_2946_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_2947_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_2948_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tanh(real),X)),aa(real,real,tanh(real),Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% tanh_real_less_iff
tff(fact_2949_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_ff(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_2950_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(real) )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_2951_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),X) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_2952_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,tanh(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% tanh_real_pos_iff
tff(fact_2953_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tanh(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% tanh_real_neg_iff
tff(fact_2954_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_2955_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_fg(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2956_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ).

% fact_mono_nat
tff(fact_2957_fact__ge__self,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),semiring_char_0_fact(nat,N))) ).

% fact_ge_self
tff(fact_2958_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,A2: real,B2: real] :
          ( ( X != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),X) = aa(A,A,real_V8093663219630862766scaleR(A,B2),X) )
           => ( A2 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_2959_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_2960_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tanh(real),X)),one_one(real))) ).

% tanh_real_lt_1
tff(fact_2961_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_2962_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C2))) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_2963_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X))) ) ) ) ).

% scaleR_right_mono
tff(fact_2964_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_2965_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_2966_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            & ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_2967_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),semiring_char_0_fact(nat,N))) ) ) ).

% dvd_fact
tff(fact_2968_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C2: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),zero_zero(real)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_2969_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,Y: A,A2: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y))) ) ) ) ).

% scaleR_left_mono
tff(fact_2970_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V2: real,A2: A,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V2)),A2) = X )
        <=> ( ( ( V2 = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V2 != zero_zero(real) )
             => ( aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V2),X) ) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_2971_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,U: real,V2: real,A2: A] :
          ( ( X = aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V2)),A2) )
        <=> ( ( ( V2 = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V2 != zero_zero(real) )
             => ( aa(A,A,real_V8093663219630862766scaleR(A,V2),X) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2) ) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_2972_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C2: A,B2: real,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E3)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),E3)),D2))) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_2973_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C2: A,B2: real,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E3)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),E3)),C2)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_2974_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N4: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),N4))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),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_2975_tanh__real__gt__neg1,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),X))) ).

% tanh_real_gt_neg1
tff(fact_2976_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
     => ( semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).

% fact_diff_Suc
tff(fact_2977_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_2978_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_2979_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_2980_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_2981_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_2982_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),X))) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_2983_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))) ) ) ).

% split_scaleR_pos_le
tff(fact_2984_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ).

% split_scaleR_neg_le
tff(fact_2985_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D2))) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_2986_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A,Y: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y))) ) ) ) ) ) ).

% scaleR_mono
tff(fact_2987_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,semiring_char_0_fact(nat,N),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),R2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R2))) ) ).

% fact_div_fact_le_pow
tff(fact_2988_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,A2: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),one_one(real)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),X)) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_2989_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),binomial(N,K)) = semiring_char_0_fact(nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_2990_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_divideR_le_eq
tff(fact_2991_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% pos_le_divideR_eq
tff(fact_2992_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% neg_divideR_le_eq
tff(fact_2993_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_le_divideR_eq
tff(fact_2994_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_less_divideR_eq
tff(fact_2995_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% neg_divideR_less_eq
tff(fact_2996_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% pos_less_divideR_eq
tff(fact_2997_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_divideR_less_eq
tff(fact_2998_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,X: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( X != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2)),aa(A,A,inverse_inverse(A),X)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_2999_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sin(real,X))) ) ) ).

% sin_gt_zero_02
tff(fact_3000_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => ( binomial(N,K) = divide_divide(nat,semiring_char_0_fact(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_3001_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb2: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb2,Xc) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa2))
         => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb2,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_3002_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,B2: nat,A2: nat,F2: fun(nat,fun(A,A)),Acc2: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A2))
       => ( set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc2) = Acc2 ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A2))
       => ( set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc2) = set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A2),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_3003_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3004_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3005_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3006_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3007_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_3008_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_3009_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_3010_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_3011_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
       => ( binomial(N,K) = zero_zero(nat) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
           => ( binomial(N,K) = binomial(N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
           => ( binomial(N,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,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),one_one(nat)),N,one_one(nat)),semiring_char_0_fact(nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_3012_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => ( suminf(A,aa(A,fun(nat,A),power_power(A),C2)) = divide_divide(A,one_one(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_3013_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fh(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3014_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_3015_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_fi(A,fun(nat,A),X),cosh(A,X)) ) ).

% cosh_converges
tff(fact_3016_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_fk(A,fun(A,fun(nat,A)),X),Y),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_3017_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_fm(A,fun(A,fun(nat,A)),X),Y),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_3018_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ? [T7: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T7))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),X))
            & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_fn(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),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),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_3019_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ? [T7: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T7))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),X))
          & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_fn(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),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),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_3020_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_fo(A,fun(nat,A),X),sinh(A,X)) ) ).

% sinh_converges
tff(fact_3021_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sinh(real,X)),sinh(real,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% sinh_real_less_iff
tff(fact_3022_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sinh(real,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% sinh_real_neg_iff
tff(fact_3023_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sinh(real,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% sinh_real_pos_iff
tff(fact_3024_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_3025_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_3026_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),pi))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,X)),cos(real,Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3027_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,X)),cos(real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3028_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y2))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y2),pi))
      & ( sin(real,Y2) = sin(real,X) )
      & ( cos(real,Y2) = cos(real,X) ) ) ).

% sincos_principal_value
tff(fact_3029_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,Y)),cos(real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3030_pi__gt__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),pi)) ).

% pi_gt_zero
tff(fact_3031_pi__not__less__zero,axiom,
    ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),zero_zero(real))) ).

% pi_not_less_zero
tff(fact_3032_sin__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),pi))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sin(real,X))) ) ) ).

% sin_gt_zero
tff(fact_3033_sinh__less__cosh__real,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sinh(real,X)),cosh(real,X))) ).

% sinh_less_cosh_real
tff(fact_3034_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) = one_one(A) )
         => ( sin(A,X) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_3035_cos__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),cos(real,X))) ) ) ).

% cos_gt_zero
tff(fact_3036_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),pi))
       => ( ( sin(real,X) = zero_zero(real) )
         => ( X = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3037_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),pi))
     => ( ( sin(real,X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3038_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),cos(real,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_3039_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,X)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_3040_pi__less__4,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) ).

% pi_less_4
tff(fact_3041_cos__two__less__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),zero_zero(real))) ).

% cos_two_less_zero
tff(fact_3042_pi__half__less__two,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% pi_half_less_two
tff(fact_3043_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_3044_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
     => ~ ! [T7: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T7))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
             => ( ( X = cos(real,T7) )
               => ( Y != sin(real,T7) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3045_pi__half__gt__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% pi_half_gt_zero
tff(fact_3046_sin__gt__zero2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sin(real,X))) ) ) ).

% sin_gt_zero2
tff(fact_3047_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,X)),zero_zero(real))) ) ) ).

% sin_lt_zero
tff(fact_3048_m2pi__less__pi,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_3049_arctan__ubound,axiom,
    ! [Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),arctan(Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% arctan_ubound
tff(fact_3050_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X))),one_one(real))) ) ) ).

% cos_double_less_one
tff(fact_3051_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sinh(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),exp(A,X)),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_3052_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),pi),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,X)),zero_zero(real))) ) ) ).

% sin_le_zero
tff(fact_3053_minus__pi__half__less__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),zero_zero(real))) ).

% minus_pi_half_less_zero
tff(fact_3054_sin__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,X)),zero_zero(real))) ) ) ).

% sin_less_zero
tff(fact_3055_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,X)),sin(real,Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3056_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,Y)),sin(real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_3057_arctan__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),arctan(Y)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),arctan(Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ).

% arctan_bounded
tff(fact_3058_arctan__lbound,axiom,
    ! [Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),arctan(Y))) ).

% arctan_lbound
tff(fact_3059_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_3060_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( sinh(real,aa(real,real,ln_ln(real),X)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),aa(real,real,inverse_inverse(real),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% sinh_ln_real
tff(fact_3061_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => ? [T7: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),T7))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),zero_zero(real)))
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_fp(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),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),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3062_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ? [T7: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T7))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),X))
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_fp(real,fun(nat,real),X)),set_ord_lessThan(nat,N))),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),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_3063_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,tan(A),X)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ) ).

% tan_double
tff(fact_3064_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( sin(real,X) = divide_divide(real,aa(real,real,tan(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% sin_tan
tff(fact_3065_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( cos(real,X) = divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% cos_tan
tff(fact_3066_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_3067_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3068_tan__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,tan(real),X))) ) ) ).

% tan_gt_zero
tff(fact_3069_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ? [X4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X4))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(real,real,tan(real),X4))) ) ) ).

% lemma_tan_total
tff(fact_3070_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X4))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
      & ( aa(real,real,tan(real),X4) = Y ) ) ).

% lemma_tan_total1
tff(fact_3071_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3072_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3073_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),X))) ) ) ) ).

% tan_monotone
tff(fact_3074_tan__total,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X4))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
      & ( aa(real,real,tan(real),X4) = Y )
      & ! [Y3: real] :
          ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
            & ( aa(real,real,tan(real),Y3) = Y ) )
         => ( Y3 = X4 ) ) ) ).

% tan_total
tff(fact_3075_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3076_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),X))) ) ) ).

% tan_pos_pi2_le
tff(fact_3077_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ? [X4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X4))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
          & ( aa(real,real,tan(real),X4) = Y ) ) ) ).

% tan_total_pos
tff(fact_3078_tan__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),X)),zero_zero(real))) ) ) ).

% tan_less_zero
tff(fact_3079_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3080_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y))) ) ) ) ).

% tan_mono_le
tff(fact_3081_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),X))),one_one(real))) ) ).

% tan_bound_pi2
tff(fact_3082_arctan,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),arctan(Y)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),arctan(Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
      & ( aa(real,real,tan(real),arctan(Y)) = Y ) ) ).

% arctan
tff(fact_3083_arctan__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( arctan(aa(real,real,tan(real),X)) = X ) ) ) ).

% arctan_tan
tff(fact_3084_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( ( aa(real,real,tan(real),X) = Y )
         => ( arctan(Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_3085_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3086_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3087_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))) = divide_divide(A,cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3088_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ? [Z3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))),Z3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))))
          & ( aa(real,real,tan(real),Z3) = X ) ) ) ).

% tan_total_pi4
tff(fact_3089_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),cos(A,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ) ).

% tan_sec
tff(fact_3090_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T7: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T7))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
             => ( Z != complex2(cos(real,T7),sin(real,T7)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3091_cot__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cot(real),X)),zero_zero(real))) ) ) ).

% cot_less_zero
tff(fact_3092_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)))),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)),one_one(nat)))))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),F2),set_ord_lessThan(nat,K))),suminf(real,F2))) ) ) ).

% sum_pos_lt_pair
tff(fact_3093_arctan__def,axiom,
    ! [Y: real] : arctan(Y) = the(real,aTP_Lamp_fq(real,fun(real,bool),Y)) ).

% arctan_def
tff(fact_3094_cot__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,cot(real),X))) ) ) ).

% cot_gt_zero
tff(fact_3095_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_3096_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_ed(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_3097_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_ec(nat,A)) ) ).

% summable_zero
tff(fact_3098_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eg(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_3099_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fr(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_3100_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),A3))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_el(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)) ) ) ).

% summable_If_finite_set
tff(fact_3101_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,bool),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),P)))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ek(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_3102_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,aa(A,fun(nat,A),power_power(A),C2))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))) ) ) ).

% summable_geometric_iff
tff(fact_3103_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] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N2))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(nat,real,G,N2))) )
           => summable(A,F2) ) ) ) ).

% summable_comparison_test'
tff(fact_3104_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(nat,real,G,N2))) )
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test
tff(fact_3105_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_fs(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_3106_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_dx(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X)))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_ft(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_3107_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,G,N2)))
         => ( summable(A,F2)
           => ( summable(A,G)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),suminf(A,F2)),suminf(A,G))) ) ) ) ) ).

% suminf_le
tff(fact_3108_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N4: set(nat),F2: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),N4))
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N4))
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_3109_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eg(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_3110_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aa(A,fun(nat,A),power_power(A),zero_zero(A))) ) ).

% summable_zero_power
tff(fact_3111_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N3: nat] : aa(nat,A,F2,N3) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3112_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),suminf(A,F2))) ) ) ) ).

% suminf_nonneg
tff(fact_3113_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F2))) ) ) ) ).

% suminf_pos
tff(fact_3114_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_de(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_3115_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_fu(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_3116_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),aa(nat,real,G,N2))) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_fv(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3117_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F2,N2))),aa(nat,real,G,N2))) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_fw(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_3118_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F2)))
            <=> ? [I4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,I4))) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3119_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I: nat] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,I)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F2))) ) ) ) ) ).

% suminf_pos2
tff(fact_3120_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( summable(A,F2)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N2))),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),suminf(A,F2)),X)) ) ) ) ).

% suminf_le_const
tff(fact_3121_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fx(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X)))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_fx(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_3122_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)))
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N2))),X))
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3123_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B3: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,N2)))
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),set_ord_atMost(nat,N2))),B3))
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_3124_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => summable(A,aa(A,fun(nat,A),power_power(A),C2)) ) ) ).

% summable_geometric
tff(fact_3125_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => summable(A,aa(A,fun(nat,A),power_power(A),X)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3126_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_eh(fun(nat,A),fun(nat,A),F2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3127_the__elem__def,axiom,
    ! [A: $tType,X6: set(A)] : the_elem(A,X6) = the(A,aTP_Lamp_fy(set(A),fun(A,bool),X6)) ).

% the_elem_def
tff(fact_3128_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I6: set(nat)] :
          ( summable(A,F2)
         => ( pp(aa(set(nat),bool,finite_finite2(nat),I6))
           => ( ! [N2: nat] :
                  ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),aa(set(nat),set(nat),uminus_uminus(set(nat)),I6)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),I6)),suminf(A,F2))) ) ) ) ) ).

% sum_le_suminf
tff(fact_3129_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),N: nat] :
          ( summable(A,F2)
         => ( ! [M5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,M5))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N))),suminf(A,F2))) ) ) ) ).

% sum_less_suminf
tff(fact_3130_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_fx(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_fx(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_fz(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3131_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_fx(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_fz(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_fx(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3132_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E3: real] :
          ( summable(A,F2)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
           => ~ ! [N9: nat] :
                  ~ ! [M2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),M2))
                     => ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,M2,N5)))),E3)) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_3133_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F2: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
         => ( summable(A,F2)
           => ? [N9: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N5))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ga(fun(nat,A),fun(nat,fun(nat,A)),F2),N5)))),R2)) ) ) ) ) ).

% suminf_exist_split
tff(fact_3134_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,I2)),one_one(real)))
     => ( ! [I2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,I2)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Z))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z),one_one(real)))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_gb(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3135_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A2: fun(nat,A),M4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R2),R0))
           => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N2))),M4))
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_gc(real,fun(fun(nat,A),fun(nat,real)),R2),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_3136_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N4: nat,F2: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),one_one(real)))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N2))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,aa(nat,nat,suc,N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))))) )
           => summable(A,F2) ) ) ) ).

% summable_ratio_test
tff(fact_3137_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),N: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M5))) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),I))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F2,I)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_ord_lessThan(nat,N))),suminf(A,F2))) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3138_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_ge(fun(nat,A),fun(A,fun(nat,A)),C2),X),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gd(fun(nat,A),fun(A,fun(nat,A)),C2),X))) ) ) ).

% diffs_equiv
tff(fact_3139_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3140_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A2,N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A2,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_gf(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),one_one(A)) ) ) ) ) ).

% pochhammer_code
tff(fact_3141_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_3142_Cauchy__iff2,axiom,
    ! [X6: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X6)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M3))
         => ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N3))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,X6,M3)),aa(nat,real,X6,N3)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_3143_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A2) = A2 ) ).

% or.left_neutral
tff(fact_3144_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),zero_zero(A)) = A2 ) ).

% or.right_neutral
tff(fact_3145_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_gg(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% The_split_eq
tff(fact_3146_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_3147_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% pochhammer_Suc0
tff(fact_3148_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(4)
tff(fact_3149_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_3150_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(3)
tff(fact_3151_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_3152_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_3153_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),zero_zero(A)) = X ) ).

% bit.disj_zero_right
tff(fact_3154_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_pos
tff(fact_3155_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,M) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A2,N) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_3156_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,N) = zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_3157_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_nonneg
tff(fact_3158_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = zero_zero(A) ) ) ) ) ).

% pochhammer_0_left
tff(fact_3159_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( X = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_3160_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,N) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),N))
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_3161_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_3162_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_3163_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_3164_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3165_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_3166_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,Z: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( comm_s3205402744901411588hammer(A,Z,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% pochhammer_product
tff(fact_3167_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => ( cos(real,aa(real,real,arcsin,X)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3168_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M3))
                 => ! [N3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N3))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,M3)),aa(nat,A,X6,N3)))),E4)) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_3169_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
             => ? [M10: nat] :
                ! [M5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M5))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,M5)),aa(nat,A,X6,N2)))),E2)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI
tff(fact_3170_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
           => ? [M8: nat] :
              ! [M2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M2))
               => ! [N5: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N5))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,M2)),aa(nat,A,X6,N5)))),E3)) ) ) ) ) ) ).

% CauchyD
tff(fact_3171_floor__real__def,axiom,
    ! [X: real] : archim6421214686448440834_floor(real,X) = the(int,aTP_Lamp_gh(real,fun(int,bool),X)) ).

% floor_real_def
tff(fact_3172_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,K4: real,C2: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),K4))
         => ( ! [X4: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X4)),K4))
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_gi(fun(nat,A),fun(A,fun(nat,A)),C2),X4)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gj(A,fun(fun(nat,A),fun(nat,A)),X),C2)) ) ) ) ).

% termdiff_converges
tff(fact_3173_Suc__0__or__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% Suc_0_or_eq
tff(fact_3174_or__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% or_Suc_0_eq
tff(fact_3175_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gk(A,fun(nat,A),Z)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_3176_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_gl(rat,fun(int,bool),X)) ).

% floor_rat_def
tff(fact_3177_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_3178_VEBT_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(vEBT_VEBT),nat,size_list(vEBT_VEBT,size_size(vEBT_VEBT)),X13)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size(3)
tff(fact_3179_is__singleton__the__elem,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the_elem(A,A3)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_3180_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% or_negative_int_iff
tff(fact_3181_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_or7035219750837199246ssThan(nat,L,U))) ).

% finite_atLeastLessThan
tff(fact_3182_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or7035219750837199246ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),U)) ) ) ) ).

% atLeastLessThan_iff
tff(fact_3183_prod__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3) = zero_zero(A) )
          <=> ? [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                & ( aa(B,A,F2,X3) = zero_zero(A) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_3184_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_3185_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),I))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),I))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),N)) ) ) ) ) ).

% ivl_subset
tff(fact_3186_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A2,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_3187_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_3188_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or7035219750837199246ssThan(A,A2,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% infinite_Ico_iff
tff(fact_3189_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_3190_prod_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ~ pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = one_one(A) ) ) ) ).

% prod.infinite
tff(fact_3191_dvd__prod__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),A2: B,B2: A,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
           => ( ( B2 = aa(B,A,F2,A2) )
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3))) ) ) ) ) ).

% dvd_prod_eqI
tff(fact_3192_dvd__prodI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),A2: B,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F2,A2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3))) ) ) ) ).

% dvd_prodI
tff(fact_3193_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,N: A,M: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),N))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,N)) = set_or7035219750837199246ssThan(A,N,M) ) ) ) ).

% ivl_diff
tff(fact_3194_prod_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gm(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gm(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = one_one(A) ) ) ) ) ) ).

% prod.delta'
tff(fact_3195_prod_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gn(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gn(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = one_one(A) ) ) ) ) ) ).

% prod.delta
tff(fact_3196_is__singletonI,axiom,
    ! [A: $tType,X: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_3197_prod_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)) ) ) ) ) ).

% prod.insert
tff(fact_3198_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_3199_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_3200_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( 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,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_3201_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( 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,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_3202_sgn__rat__def,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero(rat) )
       => ( aa(rat,rat,sgn_sgn(rat),A2) = zero_zero(rat) ) )
      & ( ( A2 != zero_zero(rat) )
       => ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A2))
           => ( aa(rat,rat,sgn_sgn(rat),A2) = one_one(rat) ) )
          & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A2))
           => ( aa(rat,rat,sgn_sgn(rat),A2) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_3203_abs__rat__def,axiom,
    ! [A2: rat] :
      ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A2),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A2) = aa(rat,rat,uminus_uminus(rat),A2) ) )
      & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A2),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A2) = A2 ) ) ) ).

% abs_rat_def
tff(fact_3204_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_3205_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),X),Y))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_3206_prod_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_mult(A) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X4: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X4))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X4),D2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or7035219750837199246ssThan(B,A2,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_3207_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R2))
     => ~ ! [S4: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),S4))
           => ! [T7: rat] :
                ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),T7))
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S4),T7) ) ) ) ) ).

% obtain_pos_sum
tff(fact_3208_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_3209_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_3210_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
            <=> ( ( A2 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_3211_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_3212_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),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),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_3213_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_3214_prod_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B3: set(C),G: fun(B,fun(C,A)),R: fun(B,fun(C,bool))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(C),bool,finite_finite2(C),B3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_go(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B3),G),R)),A3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_gq(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A3),G),R)),B3) ) ) ) ) ).

% prod.swap_restrict
tff(fact_3215_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% prod.last_plus
tff(fact_3216_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_Suc_fact
tff(fact_3217_prod__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_fact
tff(fact_3218_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gr(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gt(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_3219_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.head_if
tff(fact_3220_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I2)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),aa(B,A,G,I2))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3))) ) ) ).

% prod_mono
tff(fact_3221_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3))) ) ) ).

% prod_nonneg
tff(fact_3222_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F2,X4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3))) ) ) ).

% prod_pos
tff(fact_3223_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F2,X4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3))) ) ) ).

% prod_ge_1
tff(fact_3224_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_Suc
tff(fact_3225_prod__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ? [X2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                & ( aa(B,A,F2,X2) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3) = zero_zero(A) ) ) ) ) ).

% prod_zero
tff(fact_3226_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_3227_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or7035219750837199246ssThan(A,A2,B2))) ) ) ).

% infinite_Ico
tff(fact_3228_all__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
         => pp(aa(nat,bool,P,M3)) )
    <=> ! [X3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X3)) ) ) ).

% all_nat_less_eq
tff(fact_3229_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
          & pp(aa(nat,bool,P,M3)) )
    <=> ? [X3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X3)) ) ) ).

% ex_nat_less_eq
tff(fact_3230_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aa(nat,fun(nat,nat),minus_minus(nat),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_rev
tff(fact_3231_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gu(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_3232_lessThan__atLeast0,axiom,
    ! [N: nat] : set_ord_lessThan(nat,N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).

% lessThan_atLeast0
tff(fact_3233_prod_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_cg(set(B),fun(fun(B,bool),fun(B,bool)),A3),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_gv(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A3) ) ) ) ).

% prod.inter_filter
tff(fact_3234_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_3235_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,X4)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),one_one(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)),one_one(A))) ) ) ).

% prod_le_1
tff(fact_3236_is__singletonI_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [X4: A,Y2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
             => ( X4 = Y2 ) ) )
       => is_singleton(A,A3) ) ) ).

% is_singletonI'
tff(fact_3237_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,bool)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R,one_one(A)),one_one(A)))
         => ( ! [X15: A,Y15: A,X23: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X15),X23))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R,Y15),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(A,A,aa(A,fun(A,A),times_times(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S2))
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H,X4)),aa(B,A,G,X4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2))) ) ) ) ) ) ).

% prod.related
tff(fact_3238_prod_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)) ) ) ) ) ) ).

% prod.insert_if
tff(fact_3239_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K),N)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).

% fact_split
tff(fact_3240_prod__dvd__prod__subset,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B3: set(B),A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),B3))) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_3241_prod__dvd__prod__subset2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [B3: set(B),A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F2,A4)),aa(B,A,G,A4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3))) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_3242_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S5: set(B),T5: set(C),S2: set(B),I: fun(C,B),J: fun(B,C),T3: set(C),G: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T5))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
                 => ( aa(C,B,I,aa(B,C,J,A4)) = A4 ) )
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J,A4)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T3),T5))) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T3),T5)))
                     => ( aa(B,C,J,aa(C,B,I,B4)) = B4 ) )
                 => ( ! [B4: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T3),T5)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I,B4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5))) )
                   => ( ! [A4: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S5))
                         => ( aa(B,A,G,A4) = one_one(A) ) )
                     => ( ! [B4: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T5))
                           => ( aa(C,A,H,B4) = one_one(A) ) )
                       => ( ! [A4: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S2))
                             => ( aa(C,A,H,aa(B,C,J,A4)) = aa(B,A,G,A4) ) )
                         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),H),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_3243_sum_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_add(A) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X4: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X4))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X4),D2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or7035219750837199246ssThan(B,A2,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_3244_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,semiring_1_of_nat(A),binomial(N,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gw(nat,fun(nat,fun(nat,A)),K),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_3245_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : gbinomial(A,A2,K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gx(A,fun(nat,fun(nat,A)),A2),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_3246_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A2,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_gy(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_3247_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),gbinomial(A,A2,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gy(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_3248_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : gbinomial(A,A2,K) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gz(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_3249_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P2))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_3250_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,P2: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P2))
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M,P2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,N,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_3251_size__list__estimation,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(A,nat,F2,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(list(A),nat,size_list(A,F2),Xs))) ) ) ).

% size_list_estimation
tff(fact_3252_size__list__estimation_H,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(A,nat,F2,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(list(A),nat,size_list(A,F2),Xs))) ) ) ).

% size_list_estimation'
tff(fact_3253_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,X4)),aa(A,nat,G,X4))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_list(A,F2),Xs)),aa(list(A),nat,size_list(A,G),Xs))) ) ).

% size_list_pointwise
tff(fact_3254_prod_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_ha(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3255_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_3256_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I6: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I6))
         => ( exp(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I6)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aTP_Lamp_hb(fun(A,B),fun(A,B),F2)),I6) ) ) ) ).

% exp_sum
tff(fact_3257_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N4)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_3258_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),I: A,F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),I6))
           => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F2,I)))
             => ( ! [I2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F2,I2))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I6))) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3259_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I6))
         => ( ( I6 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F2,I2))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I6))) ) ) ) ) ).

% less_1_prod
tff(fact_3260_prod_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B3: set(B),A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3261_prod_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T3: set(B),S2: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,G,X4) = one_one(A) ) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3262_prod_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T3: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,H,X4) = one_one(A) ) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3263_prod_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T3: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,G,X4) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3264_prod_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T3: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,G,X4) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T3) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3265_prod_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C3: set(B),A3: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),A3)))
                   => ( aa(B,A,G,A4) = one_one(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B3)))
                     => ( aa(B,A,H,B4) = one_one(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C3) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B3) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3266_prod_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C3: set(B),A3: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),A3)))
                   => ( aa(B,A,G,A4) = one_one(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C3),B3)))
                     => ( aa(B,A,H,B4) = one_one(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B3) )
                  <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C3) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3267_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_3268_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( 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,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3269_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),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),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3270_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_3271_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_3272_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_3273_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_3274_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_3275_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_3276_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3277_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% sum.last_plus
tff(fact_3278_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hc(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3279_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hc(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3280_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hc(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_3281_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hc(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3282_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F2: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cl(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,N)),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_3283_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),N),set_or7035219750837199246ssThan(nat,M,N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThanSuc
tff(fact_3284_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hd(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hf(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_3285_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I2)))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,I2)),aa(B,A,G,I2))) ) )
           => ( ( A3 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3))) ) ) ) ) ).

% prod_mono_strict
tff(fact_3286_even__prod__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)))
          <=> ? [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(B,A,F2,X3))) ) ) ) ) ).

% even_prod_iff
tff(fact_3287_prod_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% prod.remove
tff(fact_3288_prod_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ).

% prod.insert_remove
tff(fact_3289_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3290_is__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
     => ~ ! [X4: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_3291_is__singleton__def,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ? [X3: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_3292_prod_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_hg(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_hg(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))) ) ) ) ) ) ).

% prod.delta_remove
tff(fact_3293_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.head_if
tff(fact_3294_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hc(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% prod.atMost_shift
tff(fact_3295_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_co(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_3296_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B3: set(A),A3: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B3))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F2,B4))) )
             => ( ! [A4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A4))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B3))) ) ) ) ) ) ).

% prod_mono2
tff(fact_3297_prod__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom_divide(A)
     => ! [A3: set(B),F2: fun(B,A),A2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( aa(B,A,F2,A2) != zero_zero(A) )
           => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))) = divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3),aa(B,A,F2,A2)) ) )
              & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3) ) ) ) ) ) ) ).

% prod_diff1
tff(fact_3298_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [N7: nat] :
                ! [M3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),M3))
                 => ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M3,N3)))),E4)) ) ) ) ) ).

% summable_Cauchy
tff(fact_3299_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gu(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_3300_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( divide_divide(nat,semiring_char_0_fact(nat,M),semiring_char_0_fact(nat,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_co(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_3301_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A,K: nat] :
          ( sums(A,F2,S)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_hh(fun(nat,A),fun(nat,fun(nat,A)),F2),K),S) ) ) ) ).

% sums_group
tff(fact_3302_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ) ).

% OR_upper
tff(fact_3303_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_lessThan(nat,N)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_3304_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hi(A,fun(nat,fun(nat,A)),A2),N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3305_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hk(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_3306_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : gbinomial(A,A2,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_gz(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_3307_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_hl(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_3308_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,I2)),aa(nat,A,A2,J2))) ) )
         => ( ! [I2: nat,J2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I2))) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_hm(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_3309_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,A2,I2)),aa(nat,nat,A2,J2))) ) )
     => ( ! [I2: nat,J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,B2,J2)),aa(nat,nat,B2,I2))) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_hn(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_3310_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(vEBT_VEBT),nat,size_list(vEBT_VEBT,vEBT_size_VEBT),X13)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size_gen(1)
tff(fact_3311_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ho(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_3312_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or7035219750837199246ssThan(int,L,U))) ).

% finite_atLeastLessThan_int
tff(fact_3313_prod__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A3) = one_one(nat) )
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => ( aa(A,nat,F2,X3) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_3314_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A3)))
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3315_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_3316_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_3317_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_3318_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_3319_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).

% quotient_of_number(4)
tff(fact_3320_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or7035219750837199246ssThan(int,zero_zero(int),U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_3321_quotient__of__div,axiom,
    ! [R2: rat,N: int,D2: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),N),D2) )
     => ( R2 = divide_divide(rat,aa(int,rat,ring_1_of_int(rat),N),aa(int,rat,ring_1_of_int(rat),D2)) ) ) ).

% quotient_of_div
tff(fact_3322_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q4: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q4)) ) ).

% quotient_of_denom_pos
tff(fact_3323_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_hp(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_3324_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_hq(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_abs_code
tff(fact_3325_ln__prod,axiom,
    ! [A: $tType,I6: set(A),F2: fun(A,real)] :
      ( pp(aa(set(A),bool,finite_finite2(A),I6))
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
           => pp(aa(real,bool,aa(real,fun(real,bool),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),I6)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_hr(fun(A,real),fun(A,real),F2)),I6) ) ) ) ).

% ln_prod
tff(fact_3326_rat__less__code,axiom,
    ! [P2: rat,Q4: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),P2),Q4))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_ht(rat,fun(int,fun(int,bool)),Q4)),quotient_of(P2))) ) ).

% rat_less_code
tff(fact_3327_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_3328_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),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),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_di(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hv(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% prod.triangle_reindex_eq
tff(fact_3329_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),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),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_dm(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hv(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% prod.triangle_reindex
tff(fact_3330_quotient__of__int,axiom,
    ! [A2: int] : quotient_of(of_int(A2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),one_one(int)) ).

% quotient_of_int
tff(fact_3331_rat__plus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P2),Q4)) = 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_hx(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_3332_rat__minus__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P2),Q4)) = 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_hz(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_3333_normalize__negative,axiom,
    ! [Q4: int,P2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Q4),zero_zero(int)))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q4))) ) ) ).

% normalize_negative
tff(fact_3334_length__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),aa(list(A),list(list(A)),subseqs(A),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_subseqs
tff(fact_3335_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_3336_subseqs__refl,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs)))) ).

% subseqs_refl
tff(fact_3337_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P2: int,Q4: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q4)) ) ).

% normalize_denom_pos
tff(fact_3338_normalize__crossproduct,axiom,
    ! [Q4: int,S: int,P2: int,R2: int] :
      ( ( Q4 != zero_zero(int) )
     => ( ( S != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R2),S)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q4) ) ) ) ) ).

% normalize_crossproduct
tff(fact_3339_rat__divide__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(divide_divide(rat,P2,Q4)) = 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_ib(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_3340_rat__times__code,axiom,
    ! [P2: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q4)) = 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_id(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P2)) ).

% rat_times_code
tff(fact_3341_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = divide_divide(rat,one_one(rat),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_3342_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ! [X4: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
         => ( aa(list(A),nat,size_size(list(A)),X4) = N ) )
     => ( aa(list(A),nat,size_size(list(A)),aa(list(list(A)),list(A),concat(A),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(list(A)),nat,size_size(list(list(A))),Xs)),N) ) ) ).

% length_mul_elem
tff(fact_3343_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_ie(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_3344_even__sum__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(B),nat,finite_card(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_if(set(B),fun(fun(B,A),fun(B,bool)),A3),F2))))) ) ) ) ).

% even_sum_iff
tff(fact_3345_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = divide_divide(rat,aa(num,rat,numeral_numeral(rat),K),aa(num,rat,numeral_numeral(rat),L)) ).

% Frct_code_post(6)
tff(fact_3346_card__Collect__less__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N))) = N ).

% card_Collect_less_nat
tff(fact_3347_card__Collect__le__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ak(nat,fun(nat,bool)),N))) = aa(nat,nat,suc,N) ).

% card_Collect_le_nat
tff(fact_3348_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_3349_card_Oinfinite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_3350_card__0__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
      <=> ( A3 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_3351_card__insert__disjoint,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_3352_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_3353_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(code_integer,M,N) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),divide_divide(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M),aa(num,code_integer,numeral_numeral(code_integer),N))),modulo_modulo(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M),aa(num,code_integer,numeral_numeral(code_integer),N))) ).

% divmod_integer'_def
tff(fact_3354_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F3: fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D3: code_integer,I2: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),F3),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),D3),I2)) ).

% full_exhaustive_integer'.cases
tff(fact_3355_exhaustive__integer_H_Ocases,axiom,
    ! [X: product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F3: fun(code_integer,option(product_prod(bool,list(code_term)))),D3: code_integer,I2: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(code_integer,option(product_prod(bool,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),F3),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),D3),I2)) ).

% exhaustive_integer'.cases
tff(fact_3356_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = zero_zero(code_integer) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
           => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
           => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = one_one(code_integer) ) ) ) ) ) ).

% sgn_integer_code
tff(fact_3357_n__subsets,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( aa(set(set(A)),nat,finite_card(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(nat,fun(set(A),bool),aTP_Lamp_ig(set(A),fun(nat,fun(set(A),bool)),A3),K))) = binomial(aa(set(A),nat,finite_card(A),A3),K) ) ) ).

% n_subsets
tff(fact_3358_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [B8: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B8))
          & ( aa(set(A),nat,finite_card(A),B8) = N )
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),A3)) ) ) ).

% infinite_arbitrarily_large
tff(fact_3359_card__subset__eq,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,finite_card(A),B3) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
tff(fact_3360_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B3: set(A),A3: set(B),R2: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( ! [A4: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A3))
           => ? [B9: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B9),B3))
                & pp(aa(A,bool,aa(B,fun(A,bool),R2,A4),B9)) ) )
       => ( ! [A12: B,A23: B,B4: A] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A12),A3))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A23),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B3))
                 => ( pp(aa(A,bool,aa(B,fun(A,bool),R2,A12),B4))
                   => ( pp(aa(A,bool,aa(B,fun(A,bool),R2,A23),B4))
                     => ( A12 = A23 ) ) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A3)),aa(set(A),nat,finite_card(A),B3))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3361_card__insert__le,axiom,
    ! [A: $tType,A3: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)))) ).

% card_insert_le
tff(fact_3362_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S: set(A),T2: set(B),R: fun(A,fun(B,bool)),K: fun(B,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( pp(aa(set(B),bool,finite_finite2(B),T2))
       => ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),T2))
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_an(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S),R),X4))) = aa(B,nat,K,X4) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_ii(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T2),R)),S) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_3363_card__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_at(set(A),fun(nat,fun(list(A),bool)),A3),N))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_3364_is__singleton__altdef,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_3365_card__eq__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
    <=> ( ( A3 = bot_bot(set(A)) )
        | ~ pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% card_eq_0_iff
tff(fact_3366_card__ge__0__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3)))
     => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ).

% card_ge_0_finite
tff(fact_3367_card__Suc__eq__finite,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B7),B10) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B7),B10))
          & ( aa(set(A),nat,finite_card(A),B10) = K )
          & pp(aa(set(A),bool,finite_finite2(A),B10)) ) ) ).

% card_Suc_eq_finite
tff(fact_3368_card__insert__if,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(set(A),nat,finite_card(A),A3) ) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ) ).

% card_insert_if
tff(fact_3369_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C3: nat] :
      ( ! [G4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),G4),F4))
         => ( pp(aa(set(A),bool,finite_finite2(A),G4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G4)),C3)) ) )
     => ( pp(aa(set(A),bool,finite_finite2(A),F4))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F4)),C3)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3370_card__seteq,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B3)),aa(set(A),nat,finite_card(A),A3)))
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
tff(fact_3371_card__mono,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))) ) ) ).

% card_mono
tff(fact_3372_obtain__subset__with__card__n,axiom,
    ! [A: $tType,N: nat,S2: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),S2)))
     => ~ ! [T6: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T6),S2))
           => ( ( aa(set(A),nat,finite_card(A),T6) = N )
             => ~ pp(aa(set(A),bool,finite_finite2(A),T6)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3373_card__less__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,finite_finite2(A),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3374_card__le__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,finite_finite2(A),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3375_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% card_length
tff(fact_3376_card__1__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) )
     => ~ ! [X4: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_3377_psubset__card__mono,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))) ) ) ).

% psubset_card_mono
tff(fact_3378_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_3379_card__less,axiom,
    ! [M4: set(nat),I: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M4))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ij(set(nat),fun(nat,fun(nat,bool)),M4),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_3380_card__less__Suc,axiom,
    ! [M4: set(nat),I: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M4))
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ik(set(nat),fun(nat,fun(nat,bool)),M4),I)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ij(set(nat),fun(nat,fun(nat,bool)),M4),I))) ) ) ).

% card_less_Suc
tff(fact_3381_card__less__Suc2,axiom,
    ! [M4: set(nat),I: nat] :
      ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M4))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ik(set(nat),fun(nat,fun(nat,bool)),M4),I))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ij(set(nat),fun(nat,fun(nat,bool)),M4),I))) ) ) ).

% card_less_Suc2
tff(fact_3382_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S2: set(A),T3: set(B),R: fun(A,fun(B,bool)),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => ( pp(aa(set(B),bool,finite_finite2(B),T3))
       => ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),T3))
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_an(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S2),R),X4))) = K ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_ii(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T3),R)),S2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T3)) ) ) ) ) ).

% sum_multicount
tff(fact_3383_sum__bounded__below,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A3: set(B),K4: A,F2: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K4),aa(B,A,F2,I2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))),K4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3))) ) ) ).

% sum_bounded_below
tff(fact_3384_sum__bounded__above,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A3: set(B),F2: fun(B,A),K4: A] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),K4)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))),K4))) ) ) ).

% sum_bounded_above
tff(fact_3385_card__gt__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3)))
    <=> ( ( A3 != bot_bot(set(A)) )
        & pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% card_gt_0_iff
tff(fact_3386_card__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B7),B10) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B7),B10))
          & ( aa(set(A),nat,finite_card(A),B10) = K )
          & ( ( K = zero_zero(nat) )
           => ( B10 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_3387_card__eq__SucD,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
     => ? [B4: A,B8: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),B8) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B8))
          & ( aa(set(A),nat,finite_card(A),B8) = K )
          & ( ( K = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_3388_card__1__singleton__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X3: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_3389_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(nat,nat,suc,zero_zero(nat))))
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A3))
               => ( X3 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3390_card__le__Suc__iff,axiom,
    ! [A: $tType,N: nat,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),A3)))
    <=> ? [A7: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),B10) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),B10))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),B10)))
          & pp(aa(set(A),bool,finite_finite2(A),B10)) ) ) ).

% card_le_Suc_iff
tff(fact_3391_card__Diff1__le,axiom,
    ! [A: $tType,A3: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))) ).

% card_Diff1_le
tff(fact_3392_card__Diff__subset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ).

% card_Diff_subset
tff(fact_3393_card__psubset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3)) ) ) ) ).

% card_psubset
tff(fact_3394_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)))) ) ).

% diff_card_le_card_Diff
tff(fact_3395_card__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_au(set(A),fun(nat,fun(list(A),bool)),A3),N))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A3))),set_ord_atMost(nat,N)) ) ) ).

% card_lists_length_le
tff(fact_3396_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aw(nat,fun(A,bool),N)))),N)) ) ) ).

% card_roots_unity
tff(fact_3397_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N4)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_3398_card__sum__le__nat__sum,axiom,
    ! [S2: set(nat)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_co(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S2)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_co(nat,nat)),S2))) ).

% card_sum_le_nat_sum
tff(fact_3399_card__nth__roots,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,bool),set(complex),collect(complex),aa(nat,fun(complex,bool),aTP_Lamp_il(complex,fun(nat,fun(complex,bool)),C2),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_3400_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,bool),set(complex),collect(complex),aTP_Lamp_cs(nat,fun(complex,bool),N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_3401_card__2__iff,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X3: A,Y4: A] :
          ( ( S2 = 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),Y4),bot_bot(set(A)))) )
          & ( X3 != Y4 ) ) ) ).

% card_2_iff
tff(fact_3402_card__3__iff,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X3: A,Y4: A,Z2: A] :
          ( ( S2 = 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),Y4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z2),bot_bot(set(A))))) )
          & ( X3 != Y4 )
          & ( Y4 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_3403_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),A3)))
     => ( A3 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_3404_card_Oremove,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_3405_card_Oinsert__remove,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_3406_card__Suc__Diff1,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_3407_card__Diff1__less,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_Diff1_less
tff(fact_3408_card__Diff2__less,axiom,
    ! [A: $tType,A3: set(A),X: A,Y: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_Diff2_less
tff(fact_3409_card__Diff1__less__iff,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A3))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_3410_card__Diff__singleton__if,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_3411_card__Diff__singleton,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_3412_Frct__code__post_I2_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_3413_Frct__code__post_I1_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A2)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_3414_Frct__code__post_I7_J,axiom,
    ! [A2: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),A2)),B2)) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),B2))) ).

% Frct_code_post(7)
tff(fact_3415_Frct__code__post_I8_J,axiom,
    ! [A2: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),aa(int,int,uminus_uminus(int),B2))) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),B2))) ).

% Frct_code_post(8)
tff(fact_3416_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_3417_prod__le__power,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F2: fun(B,A),N: A,K: nat] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F2,I2)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),N)) ) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A3)),K))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N),K))) ) ) ) ) ).

% prod_le_power
tff(fact_3418_sum__bounded__above__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & semiring_1(A) )
     => ! [A3: set(B),F2: fun(B,A),K4: A] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,I2)),K4)) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))),K4))) ) ) ) ).

% sum_bounded_above_strict
tff(fact_3419_sum__bounded__above__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_field(A)
     => ! [A3: set(B),F2: fun(B,A),K4: A] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),divide_divide(A,K4,aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))))) )
         => ( pp(aa(set(B),bool,finite_finite2(B),A3))
           => ( ( A3 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),K4)) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_3420_card__insert__le__m1,axiom,
    ! [A: $tType,N: nat,Y: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_3421_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_dl(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))),N)) ) ) ) ).

% polyfun_roots_card
tff(fact_3422_prod__gen__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A),C2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_im(B,fun(fun(B,A),fun(A,fun(B,A))),A2),B2),C2)),S2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),S2)),one_one(nat)))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_im(B,fun(fun(B,A),fun(A,fun(B,A))),A2),B2),C2)),S2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,finite_card(B),S2)) ) ) ) ) ) ).

% prod_gen_delta
tff(fact_3423_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_dl(fun(nat,A),fun(nat,fun(A,bool)),C2),N))))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_dl(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))),N)) ) ) ) ) ).

% polyfun_rootbound
tff(fact_3424_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).

% Frct_code_post(4)
tff(fact_3425_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( code_integer_of_int(K) = aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( ( ( K = zero_zero(int) )
           => ( code_integer_of_int(K) = zero_zero(code_integer) ) )
          & ( ( K != zero_zero(int) )
           => ( code_integer_of_int(K) = if(code_integer,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),code_integer_of_int(divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),code_integer_of_int(divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),one_one(code_integer))) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_3426_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A3)))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_in(set(A),fun(nat,fun(list(A),bool)),A3),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_co(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_3427_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(set(A),nat,finite_card(A),A3)))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),bool),aTP_Lamp_io(nat,fun(set(A),fun(list(A),bool)),K),A3))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_co(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_3428_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs)) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),bool),aTP_Lamp_ip(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).

% set_n_lists
tff(fact_3429_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list(bool),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Bs)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(bool),nat,size_size(list(bool)),Bs)))
            & pp(aa(nat,bool,nth(bool,Bs),N)) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_3430_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,bool)) ) ) ).

% bit_0_eq
tff(fact_3431_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),zero_zero(int)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_3432_distinct__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( distinct(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(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_3433_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),zero_zero(nat)))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ).

% bit_0
tff(fact_3434_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N))
        <=> ( ( N = zero_zero(nat) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ) ).

% bit_mod_2_iff
tff(fact_3435_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_in(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_distinct_length_eq
tff(fact_3436_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),code_integer_of_int(Xa2)),code_integer_of_int(X)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Xa2),X)) ) ).

% less_integer.abs_eq
tff(fact_3437_less__integer__code_I1_J,axiom,
    ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_integer_code(1)
tff(fact_3438_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),K) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = K ) ) ) ).

% abs_integer_code
tff(fact_3439_distinct__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( distinct(A,Xs)
     => ( distinct(B,Ys)
       => distinct(product_prod(A,B),product(A,B,Xs,Ys)) ) ) ).

% distinct_product
tff(fact_3440_distinct__zipI2,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B)] :
      ( distinct(A,Ys)
     => distinct(product_prod(B,A),zip(B,A,Xs,Ys)) ) ).

% distinct_zipI2
tff(fact_3441_distinct__zipI1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( distinct(A,Xs)
     => distinct(product_prod(A,B),zip(A,B,Xs,Ys)) ) ).

% distinct_zipI1
tff(fact_3442_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => distinct(A,Xs) ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_3443_distinct__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : distinct(product_prod(nat,A),enumerate(A,N,Xs)) ).

% distinct_enumerate
tff(fact_3444_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),N))
        <=> ( N = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_3445_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_3446_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),N))
    <=> ( N = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_3447_finite__distinct__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A3 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_3448_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_take_bit_iff
tff(fact_3449_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: bool,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(bool,A,zero_neq_one_of_bool(A),B2)),N))
        <=> ( pp(B2)
            & ( N = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_3450_subseqs__distinctD,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs))))
     => ( distinct(A,Xs)
       => distinct(A,Ys) ) ) ).

% subseqs_distinctD
tff(fact_3451_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),N))) ).

% not_bit_Suc_0_numeral
tff(fact_3452_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I: nat,J: nat] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),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_3453_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
         => ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( ( I4 != J3 )
               => ( aa(nat,A,nth(A,Xs),I4) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_3454_distinct__card,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ).

% distinct_card
tff(fact_3455_card__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
     => distinct(A,Xs) ) ).

% card_distinct
tff(fact_3456_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ? [X4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),aa(list(A),nat,size_size(list(A)),Xs)))
            & ( aa(nat,A,nth(A,Xs),X4) = X )
            & ! [Y3: nat] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y3),aa(list(A),nat,size_size(list(A)),Xs)))
                  & ( aa(nat,A,nth(A,Xs),Y3) = X ) )
               => ( Y3 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_3457_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3458_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_3459_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),N: nat,Xs: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs))))
     => ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_3460_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),M2))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) )
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))))
              <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_3461_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = zero_zero(A) )
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_3462_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),I)),bot_bot(set(A))))))
       => distinct(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),A2)) ) ) ).

% distinct_list_update
tff(fact_3463_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_3464_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_3465_length__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_n_lists
tff(fact_3466_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( ! [J2: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,J2)))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),N))
          <=> ( ( ( N = zero_zero(nat) )
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) )
              & ( ( N != zero_zero(nat) )
               => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)),N)) ) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3467_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( ( ( N = zero_zero(nat) )
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) )
            & ( ( N != zero_zero(nat) )
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ) ).

% bit_rec
tff(fact_3468_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),N)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_3469_distinct__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),union(A),Xs),Ys))
    <=> distinct(A,Ys) ) ).

% distinct_union
tff(fact_3470_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( code_int_of_integer(K) = aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K))) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( ( ( K = zero_zero(code_integer) )
           => ( code_int_of_integer(K) = zero_zero(int) ) )
          & ( ( K != zero_zero(code_integer) )
           => ( code_int_of_integer(K) = aa(product_prod(code_integer,code_integer),int,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_iq(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_3471_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),divide_divide(code_integer,K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,bool,fNot,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),dvd_dvd(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),K))) ).

% bit_cut_integer_def
tff(fact_3472_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),divide_divide(code_integer,K,L)),modulo_modulo(code_integer,K,L)) ).

% divmod_integer_def
tff(fact_3473_card__Pow,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow(A,A3)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Pow
tff(fact_3474_Pow__empty,axiom,
    ! [A: $tType] : pow(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_3475_Pow__singleton__iff,axiom,
    ! [A: $tType,X6: set(A),Y5: set(A)] :
      ( ( pow(A,X6) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Y5),bot_bot(set(set(A)))) )
    <=> ( ( X6 = bot_bot(set(A)) )
        & ( Y5 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_3476_PowI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow(A,B3))) ) ).

% PowI
tff(fact_3477_Pow__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow(A,B3)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% Pow_iff
tff(fact_3478_finite__Pow__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),pow(A,A3)))
    <=> pp(aa(set(A),bool,finite_finite2(A),A3)) ) ).

% finite_Pow_iff
tff(fact_3479_Pow__bottom,axiom,
    ! [A: $tType,B3: set(A)] : pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),bot_bot(set(A))),pow(A,B3))) ).

% Pow_bottom
tff(fact_3480_Pow__not__empty,axiom,
    ! [A: $tType,A3: set(A)] : pow(A,A3) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_3481_Pow__top,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow(A,A3))) ).

% Pow_top
tff(fact_3482_Pow__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pow(A,A3)),pow(A,B3))) ) ).

% Pow_mono
tff(fact_3483_PowD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow(A,B3)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% PowD
tff(fact_3484_Pow__def,axiom,
    ! [A: $tType,A3: set(A)] : pow(A,A3) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ag(set(A),fun(set(A),bool),A3)) ).

% Pow_def
tff(fact_3485_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),code_int_of_integer(X)),code_int_of_integer(Xa2))) ) ).

% less_integer.rep_eq
tff(fact_3486_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),L))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_iff
tff(fact_3487_binomial__def,axiom,
    ! [N: nat,K: nat] : binomial(N,K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),bool),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),bool),aTP_Lamp_ir(nat,fun(nat,fun(set(nat),bool)),N),K))) ).

% binomial_def
tff(fact_3488_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),zero_zero(code_integer)),fFalse) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( code_bit_cut_integer(K) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,bool),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,bool)),product_case_prod(code_integer,code_integer,product_prod(code_integer,bool)),aTP_Lamp_is(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_3489_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,nat,code_nat_of_integer,K) = zero_zero(nat) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,nat,code_nat_of_integer,K) = 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_it(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).

% nat_of_integer_code
tff(fact_3490_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),divide_divide(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))),modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))) ).

% divmod_abs_def
tff(fact_3491_pair__leqI2,axiom,
    ! [A2: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),S),T2))
       => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_leq)) ) ) ).

% pair_leqI2
tff(fact_3492_pair__leqI1,axiom,
    ! [A2: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_leq)) ) ).

% pair_leqI1
tff(fact_3493_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
     => ( aa(code_integer,nat,code_nat_of_integer,K) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_3494_nat__of__integer__code__post_I1_J,axiom,
    aa(code_integer,nat,code_nat_of_integer,zero_zero(code_integer)) = zero_zero(nat) ).

% nat_of_integer_code_post(1)
tff(fact_3495_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_zero(code_integer),J) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% divmod_abs_code(6)
tff(fact_3496_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_zero(code_integer)) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),aa(code_integer,code_integer,abs_abs(code_integer),J)) ).

% divmod_abs_code(5)
tff(fact_3497_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
               => ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),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_iu(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( ( L = zero_zero(code_integer) )
               => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
              & ( ( L != zero_zero(code_integer) )
               => ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),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(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_3498_wmin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),X),XS))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_weak))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert2(product_prod(nat,nat)),Y),YS))),fun_min_weak)) ) ) ) ).

% wmin_insertI
tff(fact_3499_wmax__insertI,axiom,
    ! [Y: product_prod(nat,nat),YS: set(product_prod(nat,nat)),X: product_prod(nat,nat),XS: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),Y),YS))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_max_weak))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert2(product_prod(nat,nat)),X),XS)),YS)),fun_max_weak)) ) ) ) ).

% wmax_insertI
tff(fact_3500_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_3501_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_3502_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,B),X: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),aa(C,B,F2,Y)) ).

% apsnd_conv
tff(fact_3503_wmax__emptyI,axiom,
    ! [X6: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,finite_finite2(product_prod(nat,nat)),X6))
     => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),X6)),fun_max_weak)) ) ).

% wmax_emptyI
tff(fact_3504_wmin__emptyI,axiom,
    ! [X6: set(product_prod(nat,nat))] : pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),bot_bot(set(product_prod(nat,nat))))),fun_min_weak)) ).

% wmin_emptyI
tff(fact_3505_snd__eqD,axiom,
    ! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
      ( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_3506_snd__conv,axiom,
    ! [Aa: $tType,A: $tType,X1: Aa,X22: A] : aa(product_prod(Aa,A),A,product_snd(Aa,A),aa(A,product_prod(Aa,A),aa(Aa,fun(A,product_prod(Aa,A)),product_Pair(Aa,A),X1),X22)) = X22 ).

% snd_conv
tff(fact_3507_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_3508_smin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),X),XS))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_less))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_strict))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert2(product_prod(nat,nat)),Y),YS))),fun_min_strict)) ) ) ) ).

% smin_insertI
tff(fact_3509_smax__insertI,axiom,
    ! [Y: product_prod(nat,nat),Y5: set(product_prod(nat,nat)),X: product_prod(nat,nat),X6: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),Y),Y5))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_less))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),Y5)),fun_max_strict))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert2(product_prod(nat,nat)),X),X6)),Y5)),fun_max_strict)) ) ) ) ).

% smax_insertI
tff(fact_3510_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),N: nat,Xs: list(A)] :
      ( pp(aa(set(product_prod(nat,A)),bool,aa(product_prod(nat,A),fun(set(product_prod(nat,A)),bool),member(product_prod(nat,A)),P2),aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N,Xs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_3511_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
       => ( nat_prod_decode_aux(K,M) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),M)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
       => ( nat_prod_decode_aux(K,M) = nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_3512_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
         => ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa2)) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
         => ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa2),aa(nat,nat,suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_3513_prod_Ocollapse,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) = Prod ).

% prod.collapse
tff(fact_3514_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_3515_fst__eqD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
tff(fact_3516_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X22: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = X1 ).

% fst_conv
tff(fact_3517_prod_Oexhaust__sel,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).

% prod.exhaust_sel
tff(fact_3518_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,T2: product_prod(A,B)] : T2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),T2)),aa(product_prod(A,B),B,product_snd(A,B),T2)) ).

% surjective_pairing
tff(fact_3519_prod_Osplit__sel,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(C,bool),F2: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
      ( pp(aa(C,bool,P,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),Prod)))
    <=> ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
       => pp(aa(C,bool,P,aa(B,C,aa(A,fun(B,C),F2,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).

% prod.split_sel
tff(fact_3520_prod_Osplit__sel__asm,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(C,bool),F2: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
      ( pp(aa(C,bool,P,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),Prod)))
    <=> ~ ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
          & ~ pp(aa(C,bool,P,aa(B,C,aa(A,fun(B,C),F2,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).

% prod.split_sel_asm
tff(fact_3521_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R2)))) ).

% quotient_of_denom_pos'
tff(fact_3522_smin__emptyI,axiom,
    ! [X6: set(product_prod(nat,nat))] :
      ( ( X6 != bot_bot(set(product_prod(nat,nat))) )
     => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),bot_bot(set(product_prod(nat,nat))))),fun_min_strict)) ) ).

% smin_emptyI
tff(fact_3523_smax__emptyI,axiom,
    ! [Y5: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,finite_finite2(product_prod(nat,nat)),Y5))
     => ( ( Y5 != bot_bot(set(product_prod(nat,nat))) )
       => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),Y5)),fun_max_strict)) ) ) ).

% smax_emptyI
tff(fact_3524_update__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),I: nat,Xy: product_prod(A,B)] : aa(product_prod(A,B),list(product_prod(A,B)),aa(nat,fun(product_prod(A,B),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(nat,fun(product_prod(A,B),list(product_prod(A,B)))),list_update(product_prod(A,B)),zip(A,B,Xs,Ys)),I),Xy) = zip(A,B,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),aa(product_prod(A,B),A,product_fst(A,B),Xy)),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),I),aa(product_prod(A,B),B,product_snd(A,B),Xy))) ).

% update_zip
tff(fact_3525_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B),Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),P2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
    <=> ? [N3: nat] :
          ( ( aa(nat,A,nth(A,Xs),N3) = aa(product_prod(A,B),A,product_fst(A,B),P2) )
          & ( aa(nat,B,nth(B,Ys),N3) = aa(product_prod(A,B),B,product_snd(A,B),P2) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(B),nat,size_size(list(B)),Ys))) ) ) ).

% in_set_zip
tff(fact_3526_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,nat),P2: product_prod(A,B)] : basic_BNF_size_prod(A,B,F2,G,P2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,aa(product_prod(A,B),A,product_fst(A,B),P2))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P2)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_3527_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),X: A,Y: B,A2: product_prod(A,B)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
     => ( ( A2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(A,B),A,product_fst(A,B),A2)),aa(product_prod(A,B),B,product_snd(A,B),A2))) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_3528_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),P2: A,Q: fun(B,bool),Q4: B] :
      ( pp(aa(A,bool,P,P2))
     => ( pp(aa(B,bool,Q,Q4))
       => ( pp(aa(A,bool,P,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q4))))
          & pp(aa(B,bool,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q4)))) ) ) ) ).

% conjI_realizer
tff(fact_3529_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Y: A,X: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,Y),X))
     => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)))) ) ).

% exI_realizer
tff(fact_3530_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero(nat) )
       => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
      & ( ( Y != zero_zero(nat) )
       => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y))))) ) ) ) ).

% bezw.simps
tff(fact_3531_subset__CollectI,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),Q: fun(A,bool),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3))
           => ( pp(aa(A,bool,Q,X4))
             => pp(aa(A,bool,P,X4)) ) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_am(set(A),fun(fun(A,bool),fun(A,bool)),B3),Q))),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_am(set(A),fun(fun(A,bool),fun(A,bool)),A3),P)))) ) ) ).

% subset_CollectI
tff(fact_3532_subset__Collect__iff,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_am(set(A),fun(fun(A,bool),fun(A,bool)),A3),P))))
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3))
           => pp(aa(A,bool,P,X3)) ) ) ) ).

% subset_Collect_iff
tff(fact_3533_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P2)))),one_one(int)) ).

% rat_sgn_code
tff(fact_3534_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
     => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y))))) ) ) ).

% bezw_non_0
tff(fact_3535_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( ( ( Xa2 = zero_zero(nat) )
         => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
        & ( ( Xa2 != zero_zero(nat) )
         => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa2))))) ) ) ) ) ).

% bezw.elims
tff(fact_3536_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))
       => ~ ( ( ( ( Xa2 = zero_zero(nat) )
               => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa2))))) ) ) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)) ) ) ) ).

% bezw.pelims
tff(fact_3537_normalize__def,axiom,
    ! [P2: product_prod(int,int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)))
       => ( normalize(P2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P2),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)))),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P2),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)))
       => ( ( ( aa(product_prod(int,int),int,product_snd(int,int),P2) = zero_zero(int) )
           => ( normalize(P2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ) )
          & ( ( aa(product_prod(int,int),int,product_snd(int,int),P2) != zero_zero(int) )
           => ( normalize(P2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P2),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))))),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P2),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))))) ) ) ) ) ) ).

% normalize_def
tff(fact_3538_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_mint_rel,X)
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( ( pp(A4)
                   => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                  & ( ~ pp(A4)
                   => ( ( pp(B4)
                       => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                      & ( ~ pp(B4)
                       => ( Y = none(nat) ) ) ) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Leaf(A4,B4)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi3) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_3539_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_maxt_rel,X)
       => ( ! [A4: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A4,B4) )
             => ( ( ( pp(B4)
                   => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                  & ( ~ pp(B4)
                   => ( ( pp(A4)
                       => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                      & ( ~ pp(A4)
                       => ( Y = none(nat) ) ) ) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Leaf(A4,B4)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi3: nat,Ma3: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma3) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_3540_card__UNION,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),A3))
     => ( ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),A3))
           => pp(aa(set(A),bool,finite_finite2(A),X4)) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_iw(set(set(A)),int)),aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),aTP_Lamp_ix(set(set(A)),fun(set(set(A)),bool),A3)))) ) ) ) ).

% card_UNION
tff(fact_3541_gcd__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_3542_finite__Inter,axiom,
    ! [A: $tType,M4: set(set(A))] :
      ( ? [X2: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X2),M4))
          & pp(aa(set(A),bool,finite_finite2(A),X2)) )
     => pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),M4))) ) ).

% finite_Inter
tff(fact_3543_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_3544_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,X)) = X ) ) ) ).

% cSup_atLeastAtMost
tff(fact_3545_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% cSup_singleton
tff(fact_3546_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,X,Y)) = X ) ) ) ).

% Inf_atLeastAtMost
tff(fact_3547_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_3548_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,X)) = X ) ) ) ).

% cSup_atLeastLessThan
tff(fact_3549_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_3550_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% cInf_singleton
tff(fact_3551_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_3552_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,X,Y)) = X ) ) ) ).

% Inf_atLeastLessThan
tff(fact_3553_Inf__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),set_ord_atMost(A,X)) = bot_bot(A) ) ).

% Inf_atMost
tff(fact_3554_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N)))
    <=> ( ( M != zero_zero(int) )
        | ( N != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_3555_finite__Union,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),A3))
     => ( ! [M8: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),M8),A3))
           => pp(aa(set(A),bool,finite_finite2(A),M8)) )
       => pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3))) ) ) ).

% finite_Union
tff(fact_3556_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X6: set(A),A2: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4)) )
         => ( ! [Y2: A] :
                ( ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),A2)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_3557_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),X6))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X4)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_3558_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X6: set(A),A2: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),A2)) )
         => ( ! [Y2: A] :
                ( ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),Y2)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_3559_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),X6))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Z)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_3560_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X4)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ).

% cInf_greatest
tff(fact_3561_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4)) )
           => ( ! [Y2: A] :
                  ( ! [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X2)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),A2)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_3562_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).

% cInf_le_finite
tff(fact_3563_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Z))
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Z)) ) ) ) ) ).

% cInf_lessD
tff(fact_3564_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X4)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_3565_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Z)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X6)),Z)) ) ) ) ).

% cSup_least
tff(fact_3566_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),A2)) )
           => ( ! [Y2: A] :
                  ( ! [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),Y2)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3567_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).

% le_cSup_finite
tff(fact_3568_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X6: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6)))
         => ( ( X6 != bot_bot(set(A)) )
           => ~ ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X4)) ) ) ) ) ).

% less_cSupE
tff(fact_3569_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X6)))
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X4)) ) ) ) ) ).

% less_cSupD
tff(fact_3570_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A2)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A2)) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_3571_card__Union__le__sum__card,axiom,
    ! [A: $tType,U2: set(set(A))] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),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_3572_finite__UnionD,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)))
     => pp(aa(set(set(A)),bool,finite_finite2(set(A)),A3)) ) ).

% finite_UnionD
tff(fact_3573_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6)))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X3)) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_3574_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A2))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),A2)) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_3575_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X4)),A2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S2))),A2)) ) ) ) ).

% cInf_abs_ge
tff(fact_3576_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X4)),A2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S2))),A2)) ) ) ) ).

% cSup_abs_le
tff(fact_3577_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),A2)) ) ).

% gcd_le1_int
tff(fact_3578_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),B2)) ) ).

% gcd_le2_int
tff(fact_3579_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),modulo_modulo(int,X,Y)) ) ) ).

% gcd_non_0_int
tff(fact_3580_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U2: set(set(A))] :
      ( ! [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),U2))
         => pp(aa(set(A),bool,finite_finite2(A),X4)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),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_3581_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E3: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),L))),E3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S2)),L))),E3)) ) ) ) ).

% cInf_asclose
tff(fact_3582_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E3: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),L))),E3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S2)),L))),E3)) ) ) ) ).

% cSup_asclose
tff(fact_3583_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( ( S2 = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S2)) = X ) )
            & ( ( S2 != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,complete_Sup_Sup(A),S2)) ) ) ) ) ) ).

% Sup_insert_finite
tff(fact_3584_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
               => ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa2)) ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
               => ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa2),aa(nat,nat,suc,X))) ) ) )
           => ~ accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_3585_ccpo__Sup__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% ccpo_Sup_singleton
tff(fact_3586_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_3587_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_3588_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A3) = bot_bot(A) )
        <=> ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X3))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa3),X3)) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_3589_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_3590_gcd__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A2) = A2 ).

% gcd_nat.left_neutral
tff(fact_3591_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_3592_gcd__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),zero_zero(nat)) = A2 ).

% gcd_nat.right_neutral
tff(fact_3593_gcd__0__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),zero_zero(nat)) = X ).

% gcd_0_nat
tff(fact_3594_gcd__0__left__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),X) = X ).

% gcd_0_left_nat
tff(fact_3595_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_3596_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N)))
    <=> ( ( M != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_3597_Sup__bot__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A3) = bot_bot(A) )
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => ( X3 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(1)
tff(fact_3598_Sup__bot__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A3) )
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => ( X3 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(2)
tff(fact_3599_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_3600_Inf__nat__def1,axiom,
    ! [K4: set(nat)] :
      ( ( K4 != bot_bot(set(nat)) )
     => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(set(nat),nat,complete_Inf_Inf(nat),K4)),K4)) ) ).

% Inf_nat_def1
tff(fact_3601_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2)) ) ).

% gcd_le2_nat
tff(fact_3602_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2)) ) ).

% gcd_le1_nat
tff(fact_3603_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff2_nat
tff(fact_3604_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff1_nat
tff(fact_3605_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa2) = Y )
     => ( ( ( Xa2 = zero_zero(nat) )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero(nat) )
         => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa2),modulo_modulo(nat,X,Xa2)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_3606_gcd__nat_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = X ) )
      & ( ( Y != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,X,Y)) ) ) ) ).

% gcd_nat.simps
tff(fact_3607_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,X,Y)) ) ) ).

% gcd_non_0_nat
tff(fact_3608_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X4: nat,Y2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) ) ).

% bezout_nat
tff(fact_3609_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X4: nat,Y2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) )
      | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_3610_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% Sup_upper2
tff(fact_3611_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2)) ) ) ) ).

% Sup_le_iff
tff(fact_3612_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ).

% Sup_upper
tff(fact_3613_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Z)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z)) ) ) ).

% Sup_least
tff(fact_3614_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ! [A4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
             => ? [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X2)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ).

% Sup_mono
tff(fact_3615_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),X: A] :
          ( ! [Y2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) )
         => ( ! [Y2: A] :
                ( ! [Z4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z4),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),Y2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A3) = X ) ) ) ) ).

% Sup_eqI
tff(fact_3616_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S2: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2)))
        <=> ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X3)) ) ) ) ).

% less_Sup_iff
tff(fact_3617_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X4)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A3))) ) ) ).

% Inf_greatest
tff(fact_3618_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A3)))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X3)) ) ) ) ).

% le_Inf_iff
tff(fact_3619_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),V2)) ) ) ) ).

% Inf_lower2
tff(fact_3620_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),X)) ) ) ).

% Inf_lower
tff(fact_3621_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( ! [B4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B3))
             => ? [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),B4)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ).

% Inf_mono
tff(fact_3622_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),X: A] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),I2)) )
         => ( ! [Y2: A] :
                ( ! [I3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),I3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A3) = X ) ) ) ) ).

% Inf_eqI
tff(fact_3623_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S2: set(A),A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2))
        <=> ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),A2)) ) ) ) ).

% Inf_less_iff
tff(fact_3624_empty__Union__conv,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) )
    <=> ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A3))
         => ( X3 = bot_bot(set(A)) ) ) ) ).

% empty_Union_conv
tff(fact_3625_Union__empty__conv,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) = bot_bot(set(A)) )
    <=> ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A3))
         => ( X3 = bot_bot(set(A)) ) ) ) ).

% Union_empty_conv
tff(fact_3626_Union__subsetI,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] :
      ( ! [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),A3))
         => ? [Y3: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y3),B3))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Y3)) ) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3))) ) ).

% Union_subsetI
tff(fact_3627_Union__upper,axiom,
    ! [A: $tType,B3: set(A),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B3),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3))) ) ).

% Union_upper
tff(fact_3628_Union__least,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(A)] :
      ( ! [X7: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X7),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),C3)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),C3)) ) ).

% Union_least
tff(fact_3629_Inter__greatest,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(A)] :
      ( ! [X7: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X7),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),X7)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3))) ) ).

% Inter_greatest
tff(fact_3630_Inter__lower,axiom,
    ! [A: $tType,B3: set(A),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B3),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),B3)) ) ).

% Inter_lower
tff(fact_3631_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3)))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X3)) ) ) ) ) ).

% le_Sup_iff
tff(fact_3632_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),X))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4)) ) ) ) ) ).

% Inf_le_iff
tff(fact_3633_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V3)) )
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% less_eq_Sup
tff(fact_3634_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ).

% Sup_subset_mono
tff(fact_3635_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V3),U)) )
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),U)) ) ) ) ).

% Inf_less_eq
tff(fact_3636_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ).

% Inf_superset_mono
tff(fact_3637_Union__mono,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),A3),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3))) ) ).

% Union_mono
tff(fact_3638_Inter__anti__mono,axiom,
    ! [A: $tType,B3: set(set(A)),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),B3),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3))) ) ).

% Inter_anti_mono
tff(fact_3639_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_3640_Inter__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(A)] :
      ( ! [X7: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X7),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),B3)) )
     => ( ( A3 != bot_bot(set(set(A))) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),B3)) ) ) ).

% Inter_subset
tff(fact_3641_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ).

% Inf_le_Sup
tff(fact_3642_finite__subset__Union,axiom,
    ! [A: $tType,A3: set(A),B11: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11)))
       => ~ ! [F6: set(set(A))] :
              ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),F6))
             => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F6),B11))
               => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F6))) ) ) ) ) ).

% finite_subset_Union
tff(fact_3643_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa2) = Y )
     => ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))
       => ~ ( ( ( ( Xa2 = zero_zero(nat) )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa2),modulo_modulo(nat,X,Xa2)) ) ) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)) ) ) ) ).

% gcd_nat.pelims
tff(fact_3644_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(vEBT_VEBT_minNull(X))
      <=> pp(Y) )
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y)
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fFalse,fFalse)) ) )
         => ( ! [Uv2: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv2) )
               => ( ~ pp(Y)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fTrue,Uv2)) ) )
           => ( ! [Uu2: bool] :
                  ( ( X = vEBT_Leaf(Uu2,fTrue) )
                 => ( ~ pp(Y)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(Uu2,fTrue)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( pp(Y)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ~ pp(Y)
                       => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_3645_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(vEBT_VEBT_minNull(X))
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fFalse,fFalse)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_3646_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,bool)),Q: fun(nat,fun(A,fun(A,bool)))] :
      ( ? [X_12: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,zero_zero(nat)),X_12))
     => ( ! [X4: A,N2: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N2),X4))
           => ? [Y3: A] :
                ( pp(aa(A,bool,aa(nat,fun(A,bool),P,aa(nat,nat,suc,N2)),Y3))
                & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N2),X4),Y3)) ) )
       => ? [F3: fun(nat,A)] :
          ! [N5: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N5),aa(nat,A,F3,N5)))
            & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N5),aa(nat,A,F3,N5)),aa(nat,A,F3,aa(nat,nat,suc,N5)))) ) ) ) ).

% dependent_nat_choice
tff(fact_3647_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(vEBT_VEBT_minNull(X))
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ! [Uv2: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv2) )
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fTrue,Uv2)) )
         => ( ! [Uu2: bool] :
                ( ( X = vEBT_Leaf(Uu2,fTrue) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(Uu2,fTrue)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_3648_card__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K: nat] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),C3))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)))
       => ( ! [C4: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C4),C3))
             => ( aa(set(A),nat,finite_card(A),C4) = K ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C1),C3))
               => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C22),C3))
                 => ( ( C1 != C22 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(set(A)),nat,finite_card(set(A)),C3)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) ) ) ) ) ) ).

% card_partition
tff(fact_3649_finite__enumerate,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => ? [R3: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R3,set_ord_lessThan(nat,aa(set(nat),nat,finite_card(nat),S2)))
          & ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(set(nat),nat,finite_card(nat),S2)))
             => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,R3,N5)),S2)) ) ) ) ).

% finite_enumerate
tff(fact_3650_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( ( L = zero_zero(code_integer) )
           => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
          & ( ( L != zero_zero(code_integer) )
           => ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),aa(code_integer,code_integer,sgn_sgn(code_integer),K)),aa(code_integer,code_integer,sgn_sgn(code_integer),L)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),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_iy(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_3651_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,X,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_3652_Int__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% Int_iff
tff(fact_3653_IntI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ).

% IntI
tff(fact_3654_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% inf.bounded_iff
tff(fact_3655_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) ) ) ) ).

% le_inf_iff
tff(fact_3656_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).

% inf_bot_left
tff(fact_3657_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).

% inf_bot_right
tff(fact_3658_boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_right
tff(fact_3659_boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_left
tff(fact_3660_finite__Int,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),F4))
        | pp(aa(set(A),bool,finite_finite2(A),G5)) )
     => pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),F4),G5))) ) ).

% finite_Int
tff(fact_3661_Int__subset__iff,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),A3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),B3)) ) ) ).

% Int_subset_iff
tff(fact_3662_Int__insert__right__if1,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ) ) ).

% Int_insert_right_if1
tff(fact_3663_Int__insert__right__if0,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ).

% Int_insert_right_if0
tff(fact_3664_insert__inter__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: 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),A2),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ).

% insert_inter_insert
tff(fact_3665_Int__insert__left__if1,axiom,
    ! [A: $tType,A2: A,C3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ) ) ).

% Int_insert_left_if1
tff(fact_3666_Int__insert__left__if0,axiom,
    ! [A: $tType,A2: A,C3: set(A),B3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3) ) ) ).

% Int_insert_left_if0
tff(fact_3667_Pow__Int__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow(A,A3)),pow(A,B3)) ).

% Pow_Int_eq
tff(fact_3668_in__set__remove1,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( ( A2 != B2 )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),remove1(A,B2,Xs))))
      <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% in_set_remove1
tff(fact_3669_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_3670_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_3671_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),X))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_3672_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),X) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_3673_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),X)) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_3674_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: 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),A2),A3)),B3) = bot_bot(set(A)) )
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_3675_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: 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),A2),A3)),B3) )
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ) ).

% insert_disjoint(2)
tff(fact_3676_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B3: set(A),A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = bot_bot(set(A)) )
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_3677_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A3: set(A),B2: A,B3: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3)) )
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ) ).

% disjoint_insert(2)
tff(fact_3678_Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_3679_Compl__disjoint,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_3680_Compl__disjoint2,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),A3) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_3681_Diff__Compl,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ).

% Diff_Compl
tff(fact_3682_sum__of__bool__mult__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A3: set(B),P: fun(B,bool),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_iz(fun(B,bool),fun(fun(B,A),fun(B,A)),P),F2)),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(fun(B,bool),set(B),collect(B),P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_3683_sum__mult__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A3: set(B),F2: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_ja(fun(B,A),fun(fun(B,bool),fun(B,A)),F2),P)),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(fun(B,bool),set(B),collect(B),P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_3684_sum__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A3: set(B),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_jb(fun(B,bool),fun(B,A),P)),A3) = aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(fun(B,bool),set(B),collect(B),P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_3685_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)))) ) ).

% Sup_inter_less_eq
tff(fact_3686_Union__Int__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)))) ).

% Union_Int_subset
tff(fact_3687_Int__emptyI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_3688_disjoint__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3)) ) ) ).

% disjoint_iff
tff(fact_3689_Int__empty__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B3) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_3690_Int__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_3691_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),B3))
             => ( X3 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_3692_Int__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A),D6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),D6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),D6))) ) ) ).

% Int_mono
tff(fact_3693_Int__lower1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),A3)) ).

% Int_lower1
tff(fact_3694_Int__lower2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),B3)) ).

% Int_lower2
tff(fact_3695_Int__absorb1,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = B3 ) ) ).

% Int_absorb1
tff(fact_3696_Int__absorb2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = A3 ) ) ).

% Int_absorb2
tff(fact_3697_Int__greatest,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),B3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ).

% Int_greatest
tff(fact_3698_Int__Collect__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => ( pp(aa(A,bool,P,X4))
             => pp(aa(A,bool,Q,X4)) ) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,bool),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(fun(A,bool),set(A),collect(A),Q)))) ) ) ).

% Int_Collect_mono
tff(fact_3699_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2)) ) ) ).

% inf.coboundedI2
tff(fact_3700_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2)) ) ) ).

% inf.coboundedI1
tff(fact_3701_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_3702_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb_iff1
tff(fact_3703_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),B2)) ) ).

% inf.cobounded2
tff(fact_3704_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),A2)) ) ).

% inf.cobounded1
tff(fact_3705_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.order_iff
tff(fact_3706_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))) ) ) ) ).

% inf_greatest
tff(fact_3707_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))) ) ) ) ).

% inf.boundedI
tff(fact_3708_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% inf.boundedE
tff(fact_3709_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_3710_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% inf_absorb1
tff(fact_3711_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_3712_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb1
tff(fact_3713_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% le_iff_inf
tff(fact_3714_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X4: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X4),Y2)),X4))
         => ( ! [X4: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X4),Y2)),Y2))
           => ( ! [X4: A,Y2: A,Z3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Z3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),aa(A,A,aa(A,fun(A,A),F2,Y2),Z3))) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F2,X),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_3715_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% inf.orderI
tff(fact_3716_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.orderE
tff(fact_3717_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X)) ) ) ).

% le_infI2
tff(fact_3718_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X)) ) ) ).

% le_infI1
tff(fact_3719_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D2))) ) ) ) ).

% inf_mono
tff(fact_3720_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2))) ) ) ) ).

% le_infI
tff(fact_3721_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2)) ) ) ) ).

% le_infE
tff(fact_3722_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_le2
tff(fact_3723_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_le1
tff(fact_3724_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_sup_ord(1)
tff(fact_3725_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_sup_ord(2)
tff(fact_3726_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2)) ) ) ).

% inf.strict_coboundedI2
tff(fact_3727_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2)) ) ) ).

% inf.strict_coboundedI1
tff(fact_3728_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_3729_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% inf.strict_boundedE
tff(fact_3730_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_3731_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb3
tff(fact_3732_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X)) ) ) ).

% less_infI2
tff(fact_3733_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X)) ) ) ).

% less_infI1
tff(fact_3734_remove1__commute,axiom,
    ! [A: $tType,X: A,Y: A,Zs: list(A)] : remove1(A,X,remove1(A,Y,Zs)) = remove1(A,Y,remove1(A,X,Zs)) ).

% remove1_commute
tff(fact_3735_Int__left__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)) ).

% Int_left_commute
tff(fact_3736_Int__left__absorb,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ).

% Int_left_absorb
tff(fact_3737_Int__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3) ).

% Int_commute
tff(fact_3738_Int__absorb,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),A3) = A3 ).

% Int_absorb
tff(fact_3739_Int__assoc,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Int_assoc
tff(fact_3740_IntD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ).

% IntD2
tff(fact_3741_IntD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) ) ).

% IntD1
tff(fact_3742_IntE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
         => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% IntE
tff(fact_3743_Int__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_jc(set(A),fun(set(A),fun(A,bool)),A3),B3)) ).

% Int_def
tff(fact_3744_Int__Collect,axiom,
    ! [A: $tType,X: A,A3: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,bool),set(A),collect(A),P))))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
        & pp(aa(A,bool,P,X)) ) ) ).

% Int_Collect
tff(fact_3745_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(fun(A,bool),set(A),collect(A),Q)) ).

% Collect_conj_eq
tff(fact_3746_Int__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C3)) ).

% Int_Diff
tff(fact_3747_Diff__Int2,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),B3) ).

% Diff_Int2
tff(fact_3748_Diff__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ).

% Diff_Diff_Int
tff(fact_3749_Diff__Int__distrib,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),B3)) ).

% Diff_Int_distrib
tff(fact_3750_Diff__Int__distrib2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ).

% Diff_Int_distrib2
tff(fact_3751_Int__insert__left,axiom,
    ! [A: $tType,A2: A,C3: set(A),B3: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),C3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3) ) ) ) ).

% Int_insert_left
tff(fact_3752_Int__insert__right,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ) ).

% Int_insert_right
tff(fact_3753_remove1__idem,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( remove1(A,X,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_3754_notin__set__remove1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),remove1(A,Y,Xs)))) ) ).

% notin_set_remove1
tff(fact_3755_distinct__remove1,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => distinct(A,remove1(A,X,Xs)) ) ).

% distinct_remove1
tff(fact_3756_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),B2)) = bot_bot(A) ) ).

% inf_cancel_left1
tff(fact_3757_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),B2)) = bot_bot(A) ) ).

% inf_cancel_left2
tff(fact_3758_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B3)),A2) = bot_bot(A) )
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),A2) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_3759_Diff__triv,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = A3 ) ) ).

% Diff_triv
tff(fact_3760_Int__Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_3761_Union__disjoint,axiom,
    ! [A: $tType,C3: set(set(A)),A3: 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)),C3)),A3) = bot_bot(set(A)) )
    <=> ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),C3))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),A3) = bot_bot(set(A)) ) ) ) ).

% Union_disjoint
tff(fact_3762_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_3763_Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Diff_eq
tff(fact_3764_set__remove1__subset,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,X,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_remove1_subset
tff(fact_3765_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A3: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X4: B,Y2: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),Y2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X4)),aa(B,A,H,Y2))
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,H),G)),A3) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A3)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_3766_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% inf_shunt
tff(fact_3767_finite__Inf__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X4: A,Y2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),Y2)),A3)) ) )
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Inf_Inf(A),A3)),A3)) ) ) ) ) ).

% finite_Inf_in
tff(fact_3768_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_3769_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)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(4)
tff(fact_3770_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)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(2)
tff(fact_3771_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B3))) ) ).

% disjoint_eq_subset_Compl
tff(fact_3772_insert__partition,axiom,
    ! [A: $tType,X: set(A),F4: set(set(A))] :
      ( ~ pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X),F4))
     => ( ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),X),F4)))
           => ! [Xa4: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa4),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),X),F4)))
               => ( ( X4 != Xa4 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X4),Xa4) = bot_bot(set(A)) ) ) ) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F4)) = bot_bot(set(A)) ) ) ) ).

% insert_partition
tff(fact_3773_sum_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),B3: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(B),fun(B,A),aTP_Lamp_jd(fun(B,A),fun(set(B),fun(B,A)),G),B3)),A3) ) ) ) ).

% sum.inter_restrict
tff(fact_3774_prod_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),B3: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(B),fun(B,A),aTP_Lamp_je(fun(B,A),fun(set(B),fun(B,A)),G),B3)),A3) ) ) ) ).

% prod.inter_restrict
tff(fact_3775_sum_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T3: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,finite_finite2(B),S2))
           => ( ! [I2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,H,I2) = zero_zero(A) ) )
             => ( ! [I2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),T3)))
                   => ( aa(B,A,G,I2) = zero_zero(A) ) )
               => ( ! [X4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),T3)))
                     => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T3) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_3776_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,K: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ) ) ).

% Iio_Int_singleton
tff(fact_3777_sum_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),B3: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3))) ) ) ) ).

% sum.Int_Diff
tff(fact_3778_prod_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),B3: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3))) ) ) ) ).

% prod.Int_Diff
tff(fact_3779_prod_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T3: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T3))
         => ( pp(aa(set(B),bool,finite_finite2(B),S2))
           => ( ! [I2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
                 => ( aa(B,A,H,I2) = one_one(A) ) )
             => ( ! [I2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),T3)))
                   => ( aa(B,A,G,I2) = one_one(A) ) )
               => ( ! [X4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),T3)))
                     => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T3) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_3780_card__Diff__subset__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ).

% card_Diff_subset_Int
tff(fact_3781_sum_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_jf(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(fun(B,bool),set(B),collect(B),P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,bool),set(B),collect(B),P))))) ) ) ) ).

% sum.If_cases
tff(fact_3782_prod_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_jg(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(fun(B,bool),set(B),collect(B),P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,bool),set(B),collect(B),P))))) ) ) ) ).

% prod.If_cases
tff(fact_3783_length__remove1,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% length_remove1
tff(fact_3784_dvd__partition,axiom,
    ! [A: $tType,C3: set(set(A)),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)))
     => ( ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),C3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),X4))) )
       => ( ! [X4: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),C3))
             => ! [Xa4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa4),C3))
                 => ( ( X4 != Xa4 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X4),Xa4) = bot_bot(set(A)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)))) ) ) ) ).

% dvd_partition
tff(fact_3785_sum__div__partition,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: set(B),F2: fun(B,A),B2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_jh(fun(B,A),fun(A,fun(B,A)),F2),B2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_ji(fun(B,A),fun(A,fun(B,bool)),F2),B2))))),divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_jj(fun(B,A),fun(A,fun(B,bool)),F2),B2)))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_3786_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys3: list(A)] :
            ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
           => distinct(A,Ys3) )
       => ( ! [Ys3: list(A),Zs2: list(A)] :
              ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
             => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),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,aa(list(list(A)),list(A),concat(A),Xs)) ) ) ) ).

% distinct_concat
tff(fact_3787_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( strict_mono_on(A,B,F2,A3)
        <=> ! [R5: A,S7: A] :
              ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),R5),A3))
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S7),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R5),S7)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,R5)),aa(A,B,F2,S7))) ) ) ) ).

% strict_mono_on_def
tff(fact_3788_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [R3: A,S4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),R3),A3))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S4),A3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R3),S4))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,R3)),aa(A,B,F2,S4))) ) ) )
         => strict_mono_on(A,B,F2,A3) ) ) ).

% strict_mono_onI
tff(fact_3789_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A),R2: A,S: A] :
          ( strict_mono_on(A,B,F2,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),R2),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R2),S))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,R2)),aa(A,B,F2,S))) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_3790_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F2: fun(A,B),A3: set(A),X: A,Y: A] :
          ( strict_mono_on(A,B,F2,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y))) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_3791_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_3792_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_jk(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_3793_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_jl(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_3794_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B)),X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),S2)),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R),S2))) ) ).

% inf_Int_eq2
tff(fact_3795_inf__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ).

% inf_set_def
tff(fact_3796_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_jl(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_3797_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_jk(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_3798_prod_OUnion__comp,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B3: set(set(B)),G: fun(B,A)] :
          ( ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X4),B3))
             => pp(aa(set(B),bool,finite_finite2(B),X4)) )
         => ( ! [A13: set(B)] :
                ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A13),B3))
               => ! [A24: set(B)] :
                    ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A24),B3))
                   => ( ( A13 != A24 )
                     => ! [X4: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A13))
                         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A24))
                           => ( aa(B,A,G,X4) = one_one(A) ) ) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B3)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7121269368397514597t_prod(set(B),A)),groups7121269368397514597t_prod(B,A)),G),B3) ) ) ) ) ).

% prod.Union_comp
tff(fact_3799_prod_OUnion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C3: set(set(B)),G: fun(B,A)] :
          ( ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X4),C3))
             => pp(aa(set(B),bool,finite_finite2(B),X4)) )
         => ( ! [X4: set(B)] :
                ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X4),C3))
               => ! [Xa4: set(B)] :
                    ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),Xa4),C3))
                   => ( ( X4 != Xa4 )
                     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X4),Xa4) = bot_bot(set(B)) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C3)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7121269368397514597t_prod(set(B),A)),groups7121269368397514597t_prod(B,A)),G),C3) ) ) ) ) ).

% prod.Union_disjoint
tff(fact_3800_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_3801_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_3802_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_3803_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_3804_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_3805_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_3806_sum_OUnion__comp,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B3: set(set(B)),G: fun(B,A)] :
          ( ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X4),B3))
             => pp(aa(set(B),bool,finite_finite2(B),X4)) )
         => ( ! [A13: set(B)] :
                ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A13),B3))
               => ! [A24: set(B)] :
                    ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A24),B3))
                   => ( ( A13 != A24 )
                     => ! [X4: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A13))
                         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A24))
                           => ( aa(B,A,G,X4) = zero_zero(A) ) ) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B3)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7311177749621191930dd_sum(set(B),A)),groups7311177749621191930dd_sum(B,A)),G),B3) ) ) ) ) ).

% sum.Union_comp
tff(fact_3807_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_jm(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_3808_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_jm(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_3809_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_jm(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_3810_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_jm(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_3811_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_3812_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_3813_sum_OUnion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C3: set(set(B)),G: fun(B,A)] :
          ( ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X4),C3))
             => pp(aa(set(B),bool,finite_finite2(B),X4)) )
         => ( ! [X4: set(B)] :
                ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X4),C3))
               => ! [Xa4: set(B)] :
                    ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),Xa4),C3))
                   => ( ( X4 != Xa4 )
                     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X4),Xa4) = bot_bot(set(B)) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C3)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7311177749621191930dd_sum(set(B),A)),groups7311177749621191930dd_sum(B,A)),G),C3) ) ) ) ) ).

% sum.Union_disjoint
tff(fact_3814_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A),comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_jn(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_3815_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A),comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_jo(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_3816_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys)) = binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_3817_distinct__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ! [X4: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X4),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))
         => distinct(A,X4) )
     => distinct(list(A),aa(list(list(A)),list(list(A)),product_lists(A),Xss)) ) ).

% distinct_product_lists
tff(fact_3818_finite__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,finite_finite2(list(A)),shuffles(A,Xs,Ys))) ).

% finite_shuffles
tff(fact_3819_shuffles__commutes,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : shuffles(A,Xs,Ys) = shuffles(A,Ys,Xs) ).

% shuffles_commutes
tff(fact_3820_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => ( aa(list(A),nat,size_size(list(A)),Zs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ) ) ).

% length_shuffles
tff(fact_3821_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( distinct(A,Xs)
     => ( distinct(A,Ys)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
         => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
           => distinct(A,Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_3822_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(list(A)),list(list(A)),product_lists(A),Xss))))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Xss) ) ) ).

% in_set_product_lists_length
tff(fact_3823_fstI,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),Y: A,Z: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z) )
     => ( aa(product_prod(A,B),A,product_fst(A,B),X) = Y ) ) ).

% fstI
tff(fact_3824_sndI,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),Y: A,Z: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z) )
     => ( aa(product_prod(A,B),B,product_snd(A,B),X) = Z ) ) ).

% sndI
tff(fact_3825_nth__rotate1,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate1(A),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_3826_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3827_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( bit_se4197421643247451524op_bit(A,N,A2) = A2 ) )
          & ( ( N != zero_zero(nat) )
           => ( bit_se4197421643247451524op_bit(A,N,A2) = bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).

% drop_bit_rec
tff(fact_3828_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( aa(real,real,root(N),X) = powr(real,X,divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% root_powr_inverse
tff(fact_3829_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se4197421643247451524op_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% drop_bit_negative_int_iff
tff(fact_3830_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se4197421643247451524op_bit(A,N,zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_3831_set__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rotate1(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rotate1
tff(fact_3832_length__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate1(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate1
tff(fact_3833_distinct1__rotate,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),rotate1(A),Xs))
    <=> distinct(A,Xs) ) ).

% distinct1_rotate
tff(fact_3834_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( real_Vector_of_real(A,X) = zero_zero(A) )
        <=> ( X = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_3835_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_3836_real__root__Suc__0,axiom,
    ! [X: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),X) = X ).

% real_root_Suc_0
tff(fact_3837_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = aa(real,real,root(N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_3838_root__0,axiom,
    ! [X: real] : aa(real,real,root(zero_zero(nat)),X) = zero_zero(real) ).

% root_0
tff(fact_3839_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,B2: bool] : bit_se4197421643247451524op_bit(A,N,aa(bool,A,zero_neq_one_of_bool(A),B2)) = aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat)),B2)) ) ).

% drop_bit_of_bool
tff(fact_3840_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : bit_se4197421643247451524op_bit(nat,N,aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ).

% drop_bit_of_Suc_0
tff(fact_3841_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_3842_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ).

% real_root_less_iff
tff(fact_3843_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ).

% real_root_le_iff
tff(fact_3844_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = one_one(real) )
      <=> ( X = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_3845_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_3846_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_3847_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se4197421643247451524op_bit(A,N,one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% drop_bit_of_1
tff(fact_3848_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( aa(list(A),list(A),rotate1(A),Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_3849_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ) ).

% real_root_lt_0_iff
tff(fact_3850_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_3851_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ) ).

% real_root_le_0_iff
tff(fact_3852_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_3853_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ).

% real_root_lt_1_iff
tff(fact_3854_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_3855_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ).

% real_root_le_1_iff
tff(fact_3856_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_3857_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_3858_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = A2 )
        <=> ( bit_se4197421643247451524op_bit(A,N,A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_3859_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y))) ) ) ).

% real_root_less_mono
tff(fact_3860_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y))) ) ) ).

% real_root_le_mono
tff(fact_3861_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),K) ) ) ).

% real_root_power
tff(fact_3862_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,abs_abs(real),X)) = aa(real,real,abs_abs(real),aa(real,real,root(N),X)) ) ) ).

% real_root_abs
tff(fact_3863_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(N),X)) = aa(real,real,sgn_sgn(real),X) ) ) ).

% sgn_root
tff(fact_3864_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,real_V7770717601297561774m_norm(A,X))),one_one(A))))) ) ).

% norm_less_p1
tff(fact_3865_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_gt_zero
tff(fact_3866_real__root__strict__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N4),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_3867_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,abs_abs(real),aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_3868_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_pos_pos
tff(fact_3869_real__root__strict__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N4),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_3870_real__root__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N4),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_3871_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_3872_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N) = X )
         => ( aa(real,real,root(N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_3873_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_3874_real__root__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N4))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N4),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_3875_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),aa(real,real,root(N),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),aa(real,real,root(N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_3876_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_3877_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
       => ( aa(real,real,ln_ln(real),aa(real,real,root(N),B2)) = divide_divide(real,aa(real,real,ln_ln(real),B2),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% ln_root
tff(fact_3878_log__root,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => ( aa(real,real,log(B2),aa(real,real,root(N),A2)) = divide_divide(real,aa(real,real,log(B2),A2),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% log_root
tff(fact_3879_log__base__root,axiom,
    ! [N: nat,B2: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
       => ( aa(real,real,log(aa(real,real,root(N),B2)),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),X)) ) ) ) ).

% log_base_root
tff(fact_3880_split__root,axiom,
    ! [P: fun(real,bool),N: nat,X: real] :
      ( pp(aa(real,bool,P,aa(real,real,root(N),X)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(real,bool,P,zero_zero(real))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ! [Y4: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y4)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y4)),N)) = X )
             => pp(aa(real,bool,P,Y4)) ) ) ) ) ).

% split_root
tff(fact_3881_div__add__self1__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A2: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_3882_div__add__self2__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A2: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_3883_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_jp(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_3884_Fpow__Pow__finite,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow(A,A3)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_3885_Suc__funpow,axiom,
    ! [N: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),N),suc) = aa(nat,fun(nat,nat),plus_plus(nat),N) ).

% Suc_funpow
tff(fact_3886_funpow__0,axiom,
    ! [A: $tType,F2: fun(A,A),X: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F2),X) = X ).

% funpow_0
tff(fact_3887_funpow__swap1,axiom,
    ! [A: $tType,F2: fun(A,A),N: nat,X: A] : aa(A,A,F2,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),X)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),aa(A,A,F2,X)) ).

% funpow_swap1
tff(fact_3888_funpow__mult,axiom,
    ! [A: $tType,N: nat,M: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F2)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),F2) ).

% funpow_mult
tff(fact_3889_comp__funpow,axiom,
    ! [B: $tType,A: $tType,N: nat,F2: fun(A,A)] : aa(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A)),aa(nat,fun(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A))),compow(fun(fun(B,A),fun(B,A))),N),comp(A,A,B,F2)) = comp(A,A,B,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)) ).

% comp_funpow
tff(fact_3890_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,F2),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)) ).

% funpow.simps(2)
tff(fact_3891_funpow__Suc__right,axiom,
    ! [A: $tType,N: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)),F2) ).

% funpow_Suc_right
tff(fact_3892_funpow__add,axiom,
    ! [A: $tType,M: nat,N: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F2)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2)) ).

% funpow_add
tff(fact_3893_empty__in__Fpow,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),bot_bot(set(A))),finite_Fpow(A,A3))) ).

% empty_in_Fpow
tff(fact_3894_Fpow__not__empty,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) != bot_bot(set(set(A))) ).

% Fpow_not_empty
tff(fact_3895_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_3896_Fpow__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),finite_Fpow(A,B3))) ) ).

% Fpow_mono
tff(fact_3897_Fpow__subset__Pow,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),pow(A,A3))) ).

% Fpow_subset_Pow
tff(fact_3898_Fpow__def,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_jq(set(A),fun(set(A),bool),A3)) ).

% Fpow_def
tff(fact_3899_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_3900_relpowp__bot,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),bot_bot(fun(A,fun(A,bool)))) = bot_bot(fun(A,fun(A,bool))) ) ) ).

% relpowp_bot
tff(fact_3901_relpowp__fun__conv,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y))
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = X )
          & ( aa(nat,A,F7,N) = Y )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,F7,I4)),aa(nat,A,F7,aa(nat,nat,suc,I4)))) ) ) ) ).

% relpowp_fun_conv
tff(fact_3902_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_3903_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F2: fun(C,B),G: fun(D,A),X: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G,aa(product_prod(D,C),D,product_fst(D,C),X))),aa(C,B,F2,aa(product_prod(D,C),C,product_snd(D,C),X))) ).

% apsnd_apfst
tff(fact_3904_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F2: fun(C,A),X: C,Y: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,X)),Y) ).

% apfst_conv
tff(fact_3905_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F2: fun(C,A),G: fun(D,B),X: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,aa(product_prod(C,D),C,product_fst(C,D),X))),aa(D,B,G,aa(product_prod(C,D),D,product_snd(C,D),X))) ).

% apfst_apsnd
tff(fact_3906_relpowp_Osimps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),R) = fequal(A) ).

% relpowp.simps(1)
tff(fact_3907_relpowp__0__E,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P),X),Y))
     => ( X = Y ) ) ).

% relpowp_0_E
tff(fact_3908_relpowp__0__I,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A] : pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P),X),X)) ).

% relpowp_0_I
tff(fact_3909_relpowp__E,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y2: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M5),P),X),Y2))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y2),Z)) ) ) ) ) ).

% relpowp_E
tff(fact_3910_relpowp__E2,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y2: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y2))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M5),P),Y2),Z)) ) ) ) ) ).

% relpowp_E2
tff(fact_3911_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q4: product_prod(A,B),F2: fun(C,A),P2: product_prod(C,B)] :
      ( ( Q4 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),P2) )
     => ~ ! [X4: C,Y2: B] :
            ( ( P2 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X4),Y2) )
           => ( Q4 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,X4)),Y2) ) ) ) ).

% apfst_convE
tff(fact_3912_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ak(nat,fun(nat,bool)),aTP_Lamp_aj(nat,fun(nat,bool))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_3913_set__removeAll,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,X),Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_3914_push__bit__of__Suc__0,axiom,
    ! [N: nat] : bit_se4730199178511100633sh_bit(nat,N,aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% push_bit_of_Suc_0
tff(fact_3915_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se4730199178511100633sh_bit(A,N,zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_3916_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit(A,N,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_3917_removeAll__id,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),removeAll(A,X),Xs) = Xs ) ) ).

% removeAll_id
tff(fact_3918_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se4730199178511100633sh_bit(A,N,A2)))
        <=> ( ( N != zero_zero(nat) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) ) ) ) ).

% even_push_bit_iff
tff(fact_3919_distinct__removeAll,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),removeAll(A,X),Xs)) ) ).

% distinct_removeAll
tff(fact_3920_length__removeAll__less__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,X),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_removeAll_less_eq
tff(fact_3921_distinct__remove1__removeAll,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( remove1(A,X,Xs) = aa(list(A),list(A),removeAll(A,X),Xs) ) ) ).

% distinct_remove1_removeAll
tff(fact_3922_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,bit_se4730199178511100633sh_bit(nat,M,Q4)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,Q4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_3923_length__removeAll__less,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,X),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_removeAll_less
tff(fact_3924_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se4730199178511100633sh_bit(A,N,one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_3925_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_jr(nat,fun(nat,bool))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_3926_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_js(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_3927_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,aa(list(list(A)),list(A),concat(A),Xs))
    <=> ( distinct(list(A),aa(list(list(A)),list(list(A)),removeAll(list(A),nil(A)),Xs))
        & ! [Ys4: list(A)] :
            ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
           => distinct(A,Ys4) )
        & ! [Ys4: list(A),Zs3: list(A)] :
            ( ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
              & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),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_3928_min__rpair__set,axiom,
    fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_min_strict),fun_min_weak)) ).

% min_rpair_set
tff(fact_3929_max__rpair__set,axiom,
    fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_max_strict),fun_max_weak)) ).

% max_rpair_set
tff(fact_3930_eq__snd__iff,axiom,
    ! [B: $tType,A: $tType,B2: A,P2: product_prod(B,A)] :
      ( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P2) )
    <=> ? [A7: B] : P2 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B2) ) ).

% eq_snd_iff
tff(fact_3931_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se4730199178511100633sh_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% push_bit_negative_int_iff
tff(fact_3932_zip__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( zip(A,B,Xs,Ys) = nil(product_prod(A,B)) )
    <=> ( ( Xs = nil(A) )
        | ( Ys = nil(B) ) ) ) ).

% zip_eq_Nil_iff
tff(fact_3933_Nil__eq__zip__iff,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( nil(product_prod(A,B)) = zip(A,B,Xs,Ys) )
    <=> ( ( Xs = nil(A) )
        | ( Ys = nil(B) ) ) ) ).

% Nil_eq_zip_iff
tff(fact_3934_zip__Nil,axiom,
    ! [B: $tType,A: $tType,Ys: list(B)] : zip(A,B,nil(A),Ys) = nil(product_prod(A,B)) ).

% zip_Nil
tff(fact_3935_list__update__nonempty,axiom,
    ! [A: $tType,Xs: list(A),K: nat,X: A] :
      ( ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% list_update_nonempty
tff(fact_3936_concat__replicate__trivial,axiom,
    ! [A: $tType,I: nat] : aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),I),nil(A))) = nil(A) ).

% concat_replicate_trivial
tff(fact_3937_Nil__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),nil(A)),shuffles(A,Xs,Ys)))
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_in_shuffles
tff(fact_3938_enumerate__simps_I1_J,axiom,
    ! [A: $tType,N: nat] : enumerate(A,N,nil(A)) = nil(product_prod(nat,A)) ).

% enumerate_simps(1)
tff(fact_3939_rotate1__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),rotate1(A),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rotate1_is_Nil_conv
tff(fact_3940_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_3941_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_3942_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_3943_empty__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( nil(A) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_3944_replicate__empty,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) = nil(A) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_3945_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_3946_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( nil(A) = aa(list(list(A)),list(A),concat(A),Xss) )
    <=> ! [X3: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X3),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))
         => ( X3 = nil(A) ) ) ) ).

% Nil_eq_concat_conv
tff(fact_3947_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( aa(list(list(A)),list(A),concat(A),Xss) = nil(A) )
    <=> ! [X3: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X3),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))
         => ( X3 = nil(A) ) ) ) ).

% concat_eq_Nil_conv
tff(fact_3948_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_3949_shuffles_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Ys),bot_bot(set(list(A)))) ).

% shuffles.simps(1)
tff(fact_3950_shuffles_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] : shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xs),bot_bot(set(list(A)))) ).

% shuffles.simps(2)
tff(fact_3951_Nil__in__shufflesI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = nil(A) )
     => ( ( Ys = nil(A) )
       => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),nil(A)),shuffles(A,Xs,Ys))) ) ) ).

% Nil_in_shufflesI
tff(fact_3952_concat_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(list(A)),list(A),concat(A),nil(list(A))) = nil(A) ).

% concat.simps(1)
tff(fact_3953_product_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Uu: list(B)] : product(A,B,nil(A),Uu) = nil(product_prod(A,B)) ).

% product.simps(1)
tff(fact_3954_zip_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Xs: list(A)] : zip(A,B,Xs,nil(B)) = nil(product_prod(A,B)) ).

% zip.simps(1)
tff(fact_3955_list__update__code_I1_J,axiom,
    ! [A: $tType,I: nat,Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),nil(A)),I),Y) = nil(A) ).

% list_update_code(1)
tff(fact_3956_list__update_Osimps_I1_J,axiom,
    ! [A: $tType,I: nat,V2: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),nil(A)),I),V2) = nil(A) ).

% list_update.simps(1)
tff(fact_3957_distinct_Osimps_I1_J,axiom,
    ! [A: $tType] : distinct(A,nil(A)) ).

% distinct.simps(1)
tff(fact_3958_remove1_Osimps_I1_J,axiom,
    ! [A: $tType,X: A] : remove1(A,X,nil(A)) = nil(A) ).

% remove1.simps(1)
tff(fact_3959_removeAll_Osimps_I1_J,axiom,
    ! [A: $tType,X: A] : aa(list(A),list(A),removeAll(A,X),nil(A)) = nil(A) ).

% removeAll.simps(1)
tff(fact_3960_rotate1_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rotate1(A),nil(A)) = nil(A) ).

% rotate1.simps(1)
tff(fact_3961_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_3962_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_3963_replicate__0,axiom,
    ! [A: $tType,X: A] : aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),zero_zero(nat)),X) = nil(A) ).

% replicate_0
tff(fact_3964_not__listrel1__Nil,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A))),listrel1(A,R2))) ).

% not_listrel1_Nil
tff(fact_3965_not__Nil__listrel1,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xs)),listrel1(A,R2))) ).

% not_Nil_listrel1
tff(fact_3966_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : aa(list(A),nat,size_list(A,X),nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_3967_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,bool)] : find(A,Uu,nil(A)) = none(A) ).

% find.simps(1)
tff(fact_3968_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_3969_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,bit_se4730199178511100633sh_bit(int,M,K)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_3970_prod_Osize__neq,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B)] : aa(product_prod(A,B),nat,size_size(product_prod(A,B)),X) != zero_zero(nat) ).

% prod.size_neq
tff(fact_3971_sum_Osize__neq,axiom,
    ! [A: $tType,B: $tType,X: sum_sum(A,B)] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),X) != zero_zero(nat) ).

% sum.size_neq
tff(fact_3972_Pow__set_I1_J,axiom,
    ! [A: $tType] : pow(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_3973_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A2: A,P2: product_prod(A,B)] :
      ( ( A2 = aa(product_prod(A,B),A,product_fst(A,B),P2) )
    <=> ? [B7: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B7) ) ).

% eq_fst_iff
tff(fact_3974_Gcd__remove0__nat,axiom,
    ! [M4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M4))
     => ( gcd_Gcd(nat,M4) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M4),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_3975_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ju(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% times_int.abs_eq
tff(fact_3976_prod_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I6: set(B),P2: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ax(set(B),fun(fun(B,A),fun(B,bool)),I6),P2))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I6))
             => ( groups1962203154675924110t_prod(B,A,P2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I),I6)) = groups1962203154675924110t_prod(B,A,P2,I6) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),I6))
             => ( groups1962203154675924110t_prod(B,A,P2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I),I6)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P2,I)),groups1962203154675924110t_prod(B,A,P2,I6)) ) ) ) ) ) ).

% prod.insert'
tff(fact_3977_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = remove1(A,X,aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_3978_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_3979_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(set(A),list(A),linord4507533701916653071of_set(A),bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_3980_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_3981_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(list(A),set(A),set2(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) = A3 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_3982_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: fun(B,A)] : groups1962203154675924110t_prod(B,A,P2,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_3983_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(list(A),nat,size_size(list(A)),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) = aa(set(A),nat,finite_card(A),A3) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_3984_prod_Oeq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I6: set(B),P2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( groups1962203154675924110t_prod(B,A,P2,I6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P2),I6) ) ) ) ).

% prod.eq_sum
tff(fact_3985_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = nil(A) )
          <=> ( A3 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_3986_Gcd__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,B2: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ).

% Gcd_2
tff(fact_3987_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( ( gcd_Gcd(A,A3) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))))) ) ) ).

% Gcd_0_iff
tff(fact_3988_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y2: nat] : Z != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X4),Y2)) ).

% eq_Abs_Integ
tff(fact_3989_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(set(A),list(A),linord4507533701916653071of_set(A),B3) )
         => ( pp(aa(set(A),bool,finite_finite2(A),A3))
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => ( A3 = B3 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_3990_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : distinct(A,aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_3991_Gcd__in,axiom,
    ! [A3: set(nat)] :
      ( ! [A4: nat,B4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),A4),A3))
         => ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),B4),A3))
           => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B4)),A3)) ) )
     => ( ( A3 != bot_bot(set(nat)) )
       => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),gcd_Gcd(nat,A3)),A3)) ) ) ).

% Gcd_in
tff(fact_3992_prod_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I6: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( groups1962203154675924110t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_jv(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I6) = aa(A,A,aa(A,fun(A,A),times_times(A),groups1962203154675924110t_prod(B,A,G,I6)),groups1962203154675924110t_prod(B,A,H,I6)) ) ) ) ).

% prod.distrib_triv'
tff(fact_3993_zero__int__def,axiom,
    zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))) ).

% zero_int_def
tff(fact_3994_int__def,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),N) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N),zero_zero(nat))) ).

% int_def
tff(fact_3995_prod_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
               => ( aa(B,A,G,X4) = one_one(A) ) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                 => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
             => ( groups1962203154675924110t_prod(B,A,G,T3) = groups1962203154675924110t_prod(B,A,H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3996_prod_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
               => ( aa(B,A,H,I2) = one_one(A) ) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),S2))
                 => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
             => ( groups1962203154675924110t_prod(B,A,G,S2) = groups1962203154675924110t_prod(B,A,H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3997_prod_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
               => ( aa(B,A,G,X4) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G,T3) = groups1962203154675924110t_prod(B,A,G,S2) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3998_prod_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),S2)))
               => ( aa(B,A,G,X4) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G,S2) = groups1962203154675924110t_prod(B,A,G,T3) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3999_prod_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I6: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ax(set(B),fun(fun(B,A),fun(B,bool)),I6),G))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ax(set(B),fun(fun(B,A),fun(B,bool)),I6),H))))
           => ( groups1962203154675924110t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_jv(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I6) = aa(A,A,aa(A,fun(A,A),times_times(A),groups1962203154675924110t_prod(B,A,G,I6)),groups1962203154675924110t_prod(B,A,H,I6)) ) ) ) ) ).

% prod.distrib'
tff(fact_4000_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [I6: set(B),P2: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ax(set(B),fun(fun(B,A),fun(B,bool)),I6),P2))))
           => ( groups1962203154675924110t_prod(B,A,P2,I6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P2),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ax(set(B),fun(fun(B,A),fun(B,bool)),I6),P2))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ax(set(B),fun(fun(B,A),fun(B,bool)),I6),P2))))
           => ( groups1962203154675924110t_prod(B,A,P2,I6) = one_one(A) ) ) ) ) ).

% prod.G_def
tff(fact_4001_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),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_jw(nat,fun(nat,product_prod(nat,nat)))),X)) ).

% uminus_int.abs_eq
tff(fact_4002_one__int__def,axiom,
    one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).

% one_int_def
tff(fact_4003_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_jy(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_4004_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ka(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_4005_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_4006_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ke(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_4007_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(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)))) ).

% listset.simps(1)
tff(fact_4008_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_kf(nat,bool))) ) ).

% semiring_char_def
tff(fact_4009_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_4010_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_kg(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_4011_remove1__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [X: B,F2: fun(B,A),Xs: list(B)] : remove1(B,X,aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)) = Xs ) ).

% remove1_insort_key
tff(fact_4012_length__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ) ).

% length_insort
tff(fact_4013_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_4014_insort__not__Nil,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),A2: B,Xs: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),A2),Xs) != nil(B) ) ).

% insort_not_Nil
tff(fact_4015_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),Y)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),X)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),X)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_4016_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),X),Xs)) ) ).

% insort_left_comm
tff(fact_4017_insort__key__left__comm,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Y: B,Xs: list(B)] :
          ( ( aa(B,A,F2,X) != aa(B,A,F2,Y) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Y),Xs)) ) ) ) ).

% insort_key_left_comm
tff(fact_4018_set__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] : aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),aa(list(B),set(B),set2(B),Xs)) ) ).

% set_insort_key
tff(fact_4019_distinct__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] :
          ( distinct(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs))
        <=> ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(list(B),set(B),set2(B),Xs)))
            & distinct(B,Xs) ) ) ) ).

% distinct_insort
tff(fact_4020_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_4021_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ka(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_4022_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),Xa2))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_jy(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa2))) ) ).

% less_int.rep_eq
tff(fact_4023_rat__floor__lemma,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),divide_divide(int,A2,B2))),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)))
      & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A2,B2)),one_one(int))))) ) ).

% rat_floor_lemma
tff(fact_4024_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y))
       => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_kh(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_kh(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_kh(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = bot_bot(set(nat)) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4025_image__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),X: B,A3: set(B)] :
      ( ( B2 = aa(B,A,F2,X) )
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image2(B,A,F2),A3))) ) ) ).

% image_eqI
tff(fact_4026_image__ident,axiom,
    ! [A: $tType,Y5: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ki(A,A)),Y5) = Y5 ).

% image_ident
tff(fact_4027_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_4028_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F2),A3) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_4029_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),A3) = bot_bot(set(A)) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_4030_finite__imageI,axiom,
    ! [B: $tType,A: $tType,F4: set(A),H: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),image2(A,B,H),F4))) ) ).

% finite_imageI
tff(fact_4031_insert__image,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,X)),aa(set(A),set(B),image2(A,B,F2),A3)) = aa(set(A),set(B),image2(A,B,F2),A3) ) ) ).

% insert_image
tff(fact_4032_image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: B,B3: 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),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,F2,A2)),aa(set(B),set(A),image2(B,A,F2),B3)) ).

% image_insert
tff(fact_4033_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S2) = S2 ) ).

% image_add_0
tff(fact_4034_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A3: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B3),A3)) )
        <=> ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => ( aa(B,A,B3,X3) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_4035_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B3),A3)) = bot_bot(A) )
        <=> ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => ( aa(B,A,B3,X3) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_4036_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kj(B,A)),A3)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_4037_ccSUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kk(B,A)),A3)) = bot_bot(A) ) ).

% ccSUP_bot
tff(fact_4038_SUP__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kl(A,fun(B,A),F2)),A3)) = F2 ) ) ) ).

% SUP_const
tff(fact_4039_ccSUP__const,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),F2: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_km(A,fun(B,A),F2)),A3)) = F2 ) ) ) ).

% ccSUP_const
tff(fact_4040_cSUP__const,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),C2: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kn(A,fun(B,A),C2)),A3)) = C2 ) ) ) ).

% cSUP_const
tff(fact_4041_INF__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kl(A,fun(B,A),F2)),A3)) = F2 ) ) ) ).

% INF_const
tff(fact_4042_ccINF__const,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),F2: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_km(A,fun(B,A),F2)),A3)) = F2 ) ) ) ).

% ccINF_const
tff(fact_4043_cINF__const,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),C2: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kn(A,fun(B,A),C2)),A3)) = C2 ) ) ) ).

% cINF_const
tff(fact_4044_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)) = bot_bot(A) )
        <=> ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X3))
             => ? [Xa3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,Xa3)),X3)) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_4045_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_4046_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D2))
         => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4047_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D2))
         => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ko(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,divide_divide(A,A2,D2),divide_divide(A,B2,D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4048_less__rat,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2)))) ) ) ) ).

% less_rat
tff(fact_4049_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),image2(A,B,F2),A3)))
       => ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
            & ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_kp(set(A),fun(fun(A,B),fun(A,fun(A,bool))),A3),F2),X4)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_4050_image__Pow__surj,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),A3) = B3 )
     => ( aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow(B,A3)) = pow(A,B3) ) ) ).

% image_Pow_surj
tff(fact_4051_image__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B3))) ) ).

% image_mono
tff(fact_4052_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B3: set(B)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X4)),B3)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3)) ) ).

% image_subsetI
tff(fact_4053_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3)))
     => ~ ! [C7: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C7),A3))
           => ( B3 != aa(set(B),set(A),image2(B,A,F2),C7) ) ) ) ).

% subset_imageE
tff(fact_4054_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B3))
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X3)),B3)) ) ) ).

% image_subset_iff
tff(fact_4055_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3)))
    <=> ? [AA: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),AA),A3))
          & ( B3 = aa(set(B),set(A),image2(B,A,F2),AA) ) ) ) ).

% subset_image_iff
tff(fact_4056_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ! [B10: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B10),aa(set(B),set(A),image2(B,A,F2),A3)))
         => pp(aa(set(A),bool,P,B10)) )
    <=> ! [B10: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B10),A3))
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F2),B10))) ) ) ).

% all_subset_image
tff(fact_4057_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),B2: B,F2: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( ( B2 = aa(A,B,F2,X) )
       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ).

% rev_image_eqI
tff(fact_4058_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(set(B),set(A),image2(B,A,F2),A3)))
         => pp(aa(A,bool,P,X4)) )
     => ! [X2: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
         => pp(aa(A,bool,P,aa(B,A,F2,X2))) ) ) ).

% ball_imageD
tff(fact_4059_image__cong,axiom,
    ! [B: $tType,A: $tType,M4: set(A),N4: set(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( M4 = N4 )
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),N4))
           => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) )
       => ( aa(set(A),set(B),image2(A,B,F2),M4) = aa(set(A),set(B),image2(A,B,G),N4) ) ) ) ).

% image_cong
tff(fact_4060_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(set(B),set(A),image2(B,A,F2),A3)))
          & pp(aa(A,bool,P,X2)) )
     => ? [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
          & pp(aa(A,bool,P,aa(B,A,F2,X4))) ) ) ).

% bex_imageD
tff(fact_4061_image__iff,axiom,
    ! [A: $tType,B: $tType,Z: A,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(set(B),set(A),image2(B,A,F2),A3)))
    <=> ? [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
          & ( Z = aa(B,A,F2,X3) ) ) ) ).

% image_iff
tff(fact_4062_imageI,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X)),aa(set(A),set(B),image2(A,B,F2),A3))) ) ).

% imageI
tff(fact_4063_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image2(B,A,F2),A3)))
     => ~ ! [X4: B] :
            ( ( B2 = aa(B,A,F2,X4) )
           => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3)) ) ) ).

% imageE
tff(fact_4064_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),A3: set(C)] : aa(set(B),set(A),image2(B,A,F2),aa(set(C),set(B),image2(C,B,G),A3)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_kq(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A3) ).

% image_image
tff(fact_4065_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),P: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_kr(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),F2),A3),P)) = aa(set(B),set(A),image2(B,A,F2),aa(fun(B,bool),set(B),collect(B),aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_ks(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),F2),A3),P))) ).

% Compr_image_eq
tff(fact_4066_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F2: fun(A,B),B3: set(B)] :
      ( ! [X4: A] :
          ( pp(aa(A,bool,P,X4))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X4)),B3)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(fun(A,bool),set(A),collect(A),P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_4067_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B3))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow(B,A3))),pow(A,B3))) ) ).

% image_Pow_mono
tff(fact_4068_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: 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_kt(fun(B,A),fun(B,set(A)),F2)),A3)) = aa(set(B),set(A),image2(B,A,F2),A3) ).

% UNION_singleton_eq_range
tff(fact_4069_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B3))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),finite_Fpow(B,A3))),finite_Fpow(A,B3))) ) ).

% image_Fpow_mono
tff(fact_4070_SUP__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: set(C),F2: fun(B,A),G: fun(C,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),B3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),aa(C,A,G,X2))) ) )
         => ( ! [J2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J2),B3))
               => ? [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,J2)),aa(B,A,F2,X2))) ) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G),B3)) ) ) ) ) ).

% SUP_eq
tff(fact_4071_INF__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: set(C),G: fun(C,A),F2: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),B3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,X2)),aa(B,A,F2,I2))) ) )
         => ( ! [J2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J2),B3))
               => ? [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X2)),aa(C,A,G,J2))) ) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G),B3)) ) ) ) ) ).

% INF_eq
tff(fact_4072_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),A3))) ).

% zero_notin_Suc_image
tff(fact_4073_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ! [B10: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite2(A),B10))
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B10),aa(set(B),set(A),image2(B,A,F2),A3))) )
         => pp(aa(set(A),bool,P,B10)) )
    <=> ! [B10: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),B10))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B10),A3)) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F2),B10))) ) ) ).

% all_finite_subset_image
tff(fact_4074_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ? [B10: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B10))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B10),aa(set(B),set(A),image2(B,A,F2),A3)))
          & pp(aa(set(A),bool,P,B10)) )
    <=> ? [B10: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B10))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B10),A3))
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F2),B10))) ) ) ).

% ex_finite_subset_image
tff(fact_4075_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3)))
       => ? [C7: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C7),A3))
            & pp(aa(set(B),bool,finite_finite2(B),C7))
            & ( B3 = aa(set(B),set(A),image2(B,A,F2),C7) ) ) ) ) ).

% finite_subset_image
tff(fact_4076_finite__surj,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F2),A3)))
       => pp(aa(set(B),bool,finite_finite2(B),B3)) ) ) ).

% finite_surj
tff(fact_4077_SUP__eq__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I6: set(B),F2: fun(B,A),X: A] :
          ( ( I6 != bot_bot(set(B)) )
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I6))
               => ( aa(B,A,F2,I2) = X ) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),I6)) = X ) ) ) ) ).

% SUP_eq_const
tff(fact_4078_INF__eq__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I6: set(B),F2: fun(B,A),X: A] :
          ( ( I6 != bot_bot(set(B)) )
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I6))
               => ( aa(B,A,F2,I2) = X ) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),I6)) = X ) ) ) ) ).

% INF_eq_const
tff(fact_4079_finite__image__absD,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),image2(A,A,abs_abs(A)),S2)))
         => pp(aa(set(A),bool,finite_finite2(A),S2)) ) ) ).

% finite_image_absD
tff(fact_4080_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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)),A3),B3))),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),A3)),aa(set(B),set(A),image2(B,A,F2),B3)))) ).

% image_Int_subset
tff(fact_4081_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),aa(set(B),set(A),image2(B,A,F2),B3))),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3)))) ).

% image_diff_subset
tff(fact_4082_Rat__induct__pos,axiom,
    ! [P: fun(rat,bool),Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
         => pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A4),B4))) )
     => pp(aa(rat,bool,P,Q4)) ) ).

% Rat_induct_pos
tff(fact_4083_quotient__of__eq,axiom,
    ! [A2: int,B2: int,P2: int,Q4: int] :
      ( ( quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P2),Q4) = aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) ) ) ).

% quotient_of_eq
tff(fact_4084_SUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A3: set(B),U: A,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,I)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ) ).

% SUP_upper2
tff(fact_4085_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),U))
        <=> ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),U)) ) ) ) ).

% SUP_le_iff
tff(fact_4086_SUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ).

% SUP_upper
tff(fact_4087_SUP__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] :
          ( ! [X4: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,G,X4)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),A3)))) ) ) ).

% SUP_mono'
tff(fact_4088_SUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: fun(B,A),U: A] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),U)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),U)) ) ) ).

% SUP_least
tff(fact_4089_SUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: set(C),F2: fun(B,A),G: fun(C,A)] :
          ( ! [N2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N2),A3))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),B3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,N2)),aa(C,A,G,X2))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G),B3)))) ) ) ).

% SUP_mono
tff(fact_4090_SUP__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: fun(B,A),X: A] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),X)) )
         => ( ! [Y2: A] :
                ( ! [I3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I3)),Y2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_4091_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),Y))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,I)),Y)) ) ) ) ).

% SUP_lessD
tff(fact_4092_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,F2: fun(B,A),A3: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))))
        <=> ? [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,X3))) ) ) ) ).

% less_SUP_iff
tff(fact_4093_INF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),U: A,F2: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,I2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ).

% INF_greatest
tff(fact_4094_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F2: fun(B,A),A3: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))))
        <=> ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,X3))) ) ) ) ).

% le_INF_iff
tff(fact_4095_INF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A3: set(B),F2: fun(B,A),U: A] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),U)) ) ) ) ).

% INF_lower2
tff(fact_4096_INF__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] :
          ( ! [X4: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,G,X4)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),A3)))) ) ) ).

% INF_mono'
tff(fact_4097_INF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(B,A,F2,I))) ) ) ).

% INF_lower
tff(fact_4098_INF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(B),A3: set(C),F2: fun(C,A),G: fun(B,A)] :
          ( ! [M5: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),M5),B3))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F2,X2)),aa(B,A,G,M5))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),B3)))) ) ) ).

% INF_mono
tff(fact_4099_INF__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),X: A,F2: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(B,A,F2,I2))) )
         => ( ! [Y2: A] :
                ( ! [I3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),aa(B,A,F2,I3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)) = X ) ) ) ) ).

% INF_eqI
tff(fact_4100_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F2: fun(B,A),A3: set(B),I: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(B,A,F2,I))) ) ) ) ).

% less_INF_D
tff(fact_4101_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B),A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),A2))
        <=> ? [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X3)),A2)) ) ) ) ).

% INF_less_iff
tff(fact_4102_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F2: fun(nat,A),N: nat] :
      ( ( A3 = aa(set(nat),set(A),image2(nat,A,F2),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N))) )
     => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ).

% nat_seg_image_imp_finite
tff(fact_4103_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
    <=> ? [N3: nat,F7: fun(nat,A)] : A3 = aa(set(nat),set(A),image2(nat,A,F7),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N3))) ) ).

% finite_conv_nat_seg_image
tff(fact_4104_image__constant,axiom,
    ! [A: $tType,B: $tType,X: A,A3: set(A),C2: B] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),set(B),image2(A,B,aTP_Lamp_ku(B,fun(A,B),C2)),A3) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_4105_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,A3: set(B),C2: A] :
      ( ( ( A3 = bot_bot(set(B)) )
       => ( aa(set(B),set(A),image2(B,A,aTP_Lamp_kv(A,fun(B,A),C2)),A3) = bot_bot(set(A)) ) )
      & ( ( A3 != bot_bot(set(B)) )
       => ( aa(set(B),set(A),image2(B,A,aTP_Lamp_kv(A,fun(B,A),C2)),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),bot_bot(set(A))) ) ) ) ).

% image_constant_conv
tff(fact_4106_sum_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),H: fun(B,A),G: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_kx(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S2),H),G)),aa(set(B),set(C),image2(B,C,G),S2)) ) ) ) ).

% sum.image_gen
tff(fact_4107_normalize__eq,axiom,
    ! [A2: int,B2: int,P2: int,Q4: int] :
      ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P2),Q4) = aa(int,rat,aa(int,fun(int,rat),fract,A2),B2) ) ) ).

% normalize_eq
tff(fact_4108_prod_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),H: fun(B,A),G: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_ky(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S2),H),G)),aa(set(B),set(C),image2(B,C,G),S2)) ) ) ) ).

% prod.image_gen
tff(fact_4109_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),X: A] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [Y2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
           => ( aa(A,B,F2,Y2) = aa(A,B,F2,X) ) )
       => ( the_elem(B,aa(set(A),set(B),image2(A,B,F2),A3)) = aa(A,B,F2,X) ) ) ) ).

% the_elem_image_unique
tff(fact_4110_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,F2: fun(B,A),A3: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
             => ? [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),aa(B,A,F2,X3))) ) ) ) ) ).

% le_SUP_iff
tff(fact_4111_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),X))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
             => ? [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X3)),Y4)) ) ) ) ) ).

% INF_le_iff
tff(fact_4112_SUP__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I6: set(B),C2: A,F2: fun(B,A)] :
          ( ( I6 != bot_bot(set(B)) )
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(B,A,F2,I2))) )
           => ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),I6)) = C2 )
            <=> ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I6))
                 => ( aa(B,A,F2,X3) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_4113_cSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),M4: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),M4)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),M4)) ) ) ) ).

% cSUP_least
tff(fact_4114_INF__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I6: set(B),F2: fun(B,A),C2: A] :
          ( ( I6 != bot_bot(set(B)) )
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),C2)) )
           => ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),I6)) = C2 )
            <=> ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I6))
                 => ( aa(B,A,F2,X3) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4115_cINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),M: A,F2: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F2,X4))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ) ).

% cINF_greatest
tff(fact_4116_card__image__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(A),nat,finite_card(A),A3))) ) ).

% card_image_le
tff(fact_4117_SUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,G,X4))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),B3)))) ) ) ) ).

% SUP_subset_mono
tff(fact_4118_INF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(B),A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),B3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,G,X4))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),B3)))) ) ) ) ).

% INF_superset_mono
tff(fact_4119_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_4120_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),C2: A] :
          ( ( ( A3 = bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kl(A,fun(B,A),C2)),A3)) = bot_bot(A) ) )
          & ( ( A3 != bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kl(A,fun(B,A),C2)),A3)) = C2 ) ) ) ) ).

% SUP_constant
tff(fact_4121_sum_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T3: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( pp(aa(set(C),bool,finite_finite2(C),T3))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G),S2)),T3))
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_kz(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S2),G),H)),T3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2) ) ) ) ) ) ).

% sum.group
tff(fact_4122_INF__inf__const2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I6: set(B),F2: fun(B,A),X: A] :
          ( ( I6 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_la(fun(B,A),fun(A,fun(B,A)),F2),X)),I6)) = 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,F2),I6))),X) ) ) ) ).

% INF_inf_const2
tff(fact_4123_INF__inf__const1,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I6: set(B),X: A,F2: fun(B,A)] :
          ( ( I6 != bot_bot(set(B)) )
         => ( 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_lb(A,fun(fun(B,A),fun(B,A)),X),F2)),I6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),I6))) ) ) ) ).

% INF_inf_const1
tff(fact_4124_prod_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T3: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( pp(aa(set(C),bool,finite_finite2(C),T3))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G),S2)),T3))
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_lc(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S2),G),H)),T3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2) ) ) ) ) ) ).

% prod.group
tff(fact_4125_quotient__of__Fract,axiom,
    ! [A2: int,B2: int] : quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),B2)) ).

% quotient_of_Fract
tff(fact_4126_sum_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X4: B,Y2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y2),A3))
                 => ( ( X4 != Y2 )
                   => ( ( aa(B,C,H,X4) = aa(B,C,H,Y2) )
                     => ( aa(C,A,G,aa(B,C,H,X4)) = zero_zero(A) ) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(B),set(C),image2(B,C,H),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),comp(C,A,B,G),H)),A3) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_4127_INF__le__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ).

% INF_le_SUP
tff(fact_4128_prod_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X4: B,Y2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y2),A3))
                 => ( ( X4 != Y2 )
                   => ( ( aa(B,C,H,X4) = aa(B,C,H,Y2) )
                     => ( aa(C,A,G,aa(B,C,H,X4)) = one_one(A) ) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(B),set(C),image2(B,C,H),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),comp(C,A,B,G),H)),A3) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_4129_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F2),A3)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B3)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% surj_card_le
tff(fact_4130_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,X: A,Y: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( aa(set(A),set(A),image2(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),X),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_4131_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),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),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_4132_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),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),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_4133_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,N)) = 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_ord_lessThan(nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_4134_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_atMost(nat,aa(nat,nat,suc,N)) = 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_ord_atMost(nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_4135_Fract__less__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int))) ) ) ).

% Fract_less_zero_iff
tff(fact_4136_zero__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A2)) ) ) ).

% zero_less_Fract_iff
tff(fact_4137_Fract__less__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),B2)) ) ) ).

% Fract_less_one_iff
tff(fact_4138_one__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),A2)) ) ) ).

% one_less_Fract_iff
tff(fact_4139_sum__image__le,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(C),G: fun(A,B),F2: fun(C,A)] :
          ( pp(aa(set(C),bool,finite_finite2(C),I6))
         => ( ! [I2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I2),I6))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G,aa(C,A,F2,I2)))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(C),set(A),image2(C,A,F2),I6))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,A),fun(C,B),comp(A,B,C,G),F2)),I6))) ) ) ) ).

% sum_image_le
tff(fact_4140_zero__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2)) ) ) ).

% zero_le_Fract_iff
tff(fact_4141_Fract__le__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int))) ) ) ).

% Fract_le_zero_iff
tff(fact_4142_one__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A2)) ) ) ).

% one_le_Fract_iff
tff(fact_4143_Fract__le__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),B2)) ) ) ).

% Fract_le_one_iff
tff(fact_4144_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4145_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,C2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ld(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ld(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2)) ) ) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_ld(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4146_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_le(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_le(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_le(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4147_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_lf(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_lf(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_lf(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4148_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_lg(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_lg(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_lg(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4149_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_lh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_lh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,M)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_lh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4150_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S2: set(A),R: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( pp(aa(set(B),bool,finite_finite2(B),R))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S2)),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_li(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),S2) = 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_lk(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S2),G),F2)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_4151_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_ll(A,fun(nat,A),B3)),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ).

% INF_nat_binary
tff(fact_4152_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_lm(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_ln(A,set(D))) ).

% empty_natural
tff(fact_4153_uminus__int__def,axiom,
    uminus_uminus(int) = aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_jw(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_4154_sorted__key__list__of__set__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A)] : linord144544945434240204of_set(B,A,F2) = finite_folding_F(B,list(B),linorder_insort_key(B,A,F2),nil(B)) ) ).

% sorted_key_list_of_set_def
tff(fact_4155_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set(product_prod(A,B)),F2: fun(A,fun(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),A3))
     => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,aa(A,fun(B,C),F2,A2),B2)),aa(set(product_prod(A,B)),set(C),image2(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2)),A3))) ) ).

% pair_imageI
tff(fact_4156_finite__UN,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3))))
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B3,X3))) ) ) ) ).

% finite_UN
tff(fact_4157_UN__constant,axiom,
    ! [B: $tType,A: $tType,A3: set(B),C2: set(A)] :
      ( ( ( A3 = bot_bot(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_lo(set(A),fun(B,set(A)),C2)),A3)) = bot_bot(set(A)) ) )
      & ( ( A3 != bot_bot(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_lo(set(A),fun(B,set(A)),C2)),A3)) = C2 ) ) ) ).

% UN_constant
tff(fact_4158_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B3,A4))) )
       => pp(aa(set(B),bool,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3)))) ) ) ).

% finite_UN_I
tff(fact_4159_finite__INT,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A3: fun(A,set(B))] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),I6))
          & pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),A3,X2))) )
     => pp(aa(set(B),bool,finite_finite2(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I6)))) ) ).

% finite_INT
tff(fact_4160_UN__singleton,axiom,
    ! [A: $tType,A3: 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_lp(A,set(A))),A3)) = A3 ).

% UN_singleton
tff(fact_4161_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C3: set(B),A2: A,B3: fun(B,set(A))] :
      ( ( ( C3 = bot_bot(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_lq(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B3)),C3)) = bot_bot(set(A)) ) )
      & ( ( C3 != bot_bot(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_lq(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B3)),C3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) ) ) ) ).

% UN_simps(1)
tff(fact_4162_set__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),set(A),set2(A),aa(list(list(A)),list(A),concat(A),Xs)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).

% set_concat
tff(fact_4163_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,bool))),X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),S2),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),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),bool)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool)),S2))))) ) ).

% Inf_INT_eq2
tff(fact_4164_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,bool))),X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),S2),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),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),bool)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool)),S2))))) ) ).

% Sup_SUP_eq2
tff(fact_4165_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S2: set(C),X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_lr(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R2)),S2)),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),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),S2)))) ) ).

% INF_INT_eq2
tff(fact_4166_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S2: set(C),X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_lr(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R2)),S2)),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),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),S2)))) ) ).

% SUP_UN_eq2
tff(fact_4167_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool)))),S2)),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S2))) ) ).

% SUP_Sup_eq2
tff(fact_4168_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool)))),S2)),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S2))) ) ).

% INF_Int_eq2
tff(fact_4169_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_4170_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,set(A)),G: fun(C,set(B)),X: C,X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),aa(C,set(B),G,X)))),X6))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(C,set(B),G,X)),aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_ls(fun(B,set(A)),fun(set(A),fun(B,bool)),F2),X6)))) ) ).

% UN_image_subset
tff(fact_4171_None__notin__image__Some,axiom,
    ! [A: $tType,A3: set(A)] : ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),none(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A3))) ).

% None_notin_image_Some
tff(fact_4172_INF__filter__not__bot,axiom,
    ! [I7: $tType,A: $tType,B3: set(I7),F4: fun(I7,filter(A))] :
      ( ! [X7: set(I7)] :
          ( pp(aa(set(I7),bool,aa(set(I7),fun(set(I7),bool),ord_less_eq(set(I7)),X7),B3))
         => ( pp(aa(set(I7),bool,finite_finite2(I7),X7))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I7),set(filter(A)),image2(I7,filter(A),F4),X7)) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I7),set(filter(A)),image2(I7,filter(A),F4),B3)) != bot_bot(filter(A)) ) ) ).

% INF_filter_not_bot
tff(fact_4173_finite__int__iff__bounded,axiom,
    ! [S2: set(int)] :
      ( pp(aa(set(int),bool,finite_finite2(int),S2))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S2)),set_ord_lessThan(int,K3))) ) ).

% finite_int_iff_bounded
tff(fact_4174_finite__int__iff__bounded__le,axiom,
    ! [S2: set(int)] :
      ( pp(aa(set(int),bool,finite_finite2(int),S2))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S2)),set_ord_atMost(int,K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_4175_UN__empty2,axiom,
    ! [B: $tType,A: $tType,A3: 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_lt(B,set(A))),A3)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_4176_UN__empty,axiom,
    ! [B: $tType,A: $tType,B3: 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),B3),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_4177_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A3: 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),B3),A3)) )
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
         => ( aa(B,set(A),B3,X3) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_4178_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A3: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A3)) = bot_bot(set(A)) )
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
         => ( aa(B,set(A),B3,X3) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_4179_UN__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F2,X4)),aa(A,set(B),G,X4))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),A3))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),B3)))) ) ) ).

% UN_mono
tff(fact_4180_UN__least,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(A,set(B)),C3: set(B)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B3,X4)),C3)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3))),C3)) ) ).

% UN_least
tff(fact_4181_UN__upper,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B3,A2)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3)))) ) ).

% UN_upper
tff(fact_4182_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I6: set(B),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),I6))),B3))
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I6))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),A3,X3)),B3)) ) ) ).

% UN_subset_iff
tff(fact_4183_INT__lower,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3))),aa(A,set(B),B3,A2))) ) ).

% INT_lower
tff(fact_4184_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C3: set(B),B3: fun(A,set(B))] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C3),aa(A,set(B),B3,X4))) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3)))) ) ).

% INT_greatest
tff(fact_4185_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F2,X4)),aa(A,set(B),G,X4))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),B3))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),A3)))) ) ) ).

% INT_anti_mono
tff(fact_4186_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A3: fun(B,set(A)),I6: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),I6))))
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I6))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(B,set(A),A3,X3))) ) ) ).

% INT_subset_iff
tff(fact_4187_in__image__insert__iff,axiom,
    ! [A: $tType,B3: set(set(A)),X: A,A3: set(A)] :
      ( ! [C7: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C7),B3))
         => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),C7)) )
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X)),B3)))
      <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
          & pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),B3)) ) ) ) ).

% in_image_insert_iff
tff(fact_4188_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C3: set(B),A2: A,B3: fun(B,set(A))] :
      ( ( ( C3 = bot_bot(set(B)) )
       => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ) )
      & ( ( C3 != bot_bot(set(B)) )
       => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C3))) = 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_lq(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B3)),C3)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_4189_INT__extend__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C3: set(D),A3: set(C),B3: fun(D,set(C))] :
      ( ( ( C3 = bot_bot(set(D)) )
       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A3),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B3),C3))) = A3 ) )
      & ( ( C3 != bot_bot(set(D)) )
       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A3),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B3),C3))) = aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_lu(set(C),fun(fun(D,set(C)),fun(D,set(C))),A3),B3)),C3)) ) ) ) ).

% INT_extend_simps(2)
tff(fact_4190_INT__extend__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,C3: set(A),A3: fun(A,set(B)),B3: set(B)] :
      ( ( ( C3 = bot_bot(set(A)) )
       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),C3))),B3) = B3 ) )
      & ( ( C3 != bot_bot(set(A)) )
       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),C3))),B3) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_lv(fun(A,set(B)),fun(set(B),fun(A,set(B))),A3),B3)),C3)) ) ) ) ).

% INT_extend_simps(1)
tff(fact_4191_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B11: set(set(A)),A3: set(A)] :
      ( ( ( B11 = bot_bot(set(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)),B11)),A3) = A3 ) )
      & ( ( B11 != bot_bot(set(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)),B11)),A3) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_lw(set(A),fun(set(A),set(A)),A3)),B11)) ) ) ) ).

% Int_Inter_eq(2)
tff(fact_4192_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,B11: set(set(A)),A3: set(A)] :
      ( ( ( B11 = bot_bot(set(set(A))) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)) = A3 ) )
      & ( ( B11 != bot_bot(set(set(A))) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)) = 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)),inf_inf(set(A)),A3)),B11)) ) ) ) ).

% Int_Inter_eq(1)
tff(fact_4193_INT__extend__simps_I4_J,axiom,
    ! [G2: $tType,H3: $tType,C3: set(H3),A3: set(G2),B3: fun(H3,set(G2))] :
      ( ( ( C3 = bot_bot(set(H3)) )
       => ( aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),A3),aa(set(set(G2)),set(G2),complete_Sup_Sup(set(G2)),aa(set(H3),set(set(G2)),image2(H3,set(G2),B3),C3))) = A3 ) )
      & ( ( C3 != bot_bot(set(H3)) )
       => ( aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),A3),aa(set(set(G2)),set(G2),complete_Sup_Sup(set(G2)),aa(set(H3),set(set(G2)),image2(H3,set(G2),B3),C3))) = aa(set(set(G2)),set(G2),complete_Inf_Inf(set(G2)),aa(set(H3),set(set(G2)),image2(H3,set(G2),aa(fun(H3,set(G2)),fun(H3,set(G2)),aTP_Lamp_lx(set(G2),fun(fun(H3,set(G2)),fun(H3,set(G2))),A3),B3)),C3)) ) ) ) ).

% INT_extend_simps(4)
tff(fact_4194_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite2(int),S2))
    <=> ~ pp(aa(set(nat),bool,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))),S2))) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_4195_subset__subseqs,axiom,
    ! [A: $tType,X6: set(A),Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs))))) ) ).

% subset_subseqs
tff(fact_4196_subseqs__powset,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs))) = pow(A,aa(list(A),set(A),set2(A),Xs)) ).

% subseqs_powset
tff(fact_4197_conj__subset__def,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(fun(A,bool),set(A),collect(A),P)))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(fun(A,bool),set(A),collect(A),Q))) ) ) ).

% conj_subset_def
tff(fact_4198_sum_OUNION__disjoint,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I6: set(B),A3: fun(B,set(C)),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I6))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A3,X4))) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I6))
                 => ! [Xa4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa4),I6))
                     => ( ( X4 != Xa4 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A3,X4)),aa(B,set(C),A3,Xa4)) = bot_bot(set(C)) ) ) ) )
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A3),I6))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_ly(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A3),G)),I6) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_4199_prod_OUNION__disjoint,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I6: set(B),A3: fun(B,set(C)),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I6))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A3,X4))) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I6))
                 => ! [Xa4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa4),I6))
                     => ( ( X4 != Xa4 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A3,X4)),aa(B,set(C),A3,Xa4)) = bot_bot(set(C)) ) ) ) )
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A3),I6))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_lz(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A3),G)),I6) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_4200_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),I6))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I6)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ma(fun(A,set(B)),fun(A,nat),A3)),I6))) ) ).

% card_UN_le
tff(fact_4201_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),I6))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),I6))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),A3,X4))) )
       => ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),I6))
             => ! [Xa4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),I6))
                 => ( ( X4 != Xa4 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X4)),aa(A,set(B),A3,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),A3),I6))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ma(fun(A,set(B)),fun(A,nat),A3)),I6) ) ) ) ) ).

% card_UN_disjoint
tff(fact_4202_times__int__def,axiom,
    times_times(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ju(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_4203_minus__int__def,axiom,
    minus_minus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ke(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_4204_plus__int__def,axiom,
    plus_plus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_4205_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( 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),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_4206_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I6: set(A),F4: fun(A,filter(B))] :
      ( ! [I2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
         => ! [J2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I6))
             => ? [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),I6))
                  & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F4,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I2)),aa(A,filter(B),F4,J2)))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I6)) = bot_bot(filter(B)) )
      <=> ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),I6))
            & ( aa(A,filter(B),F4,X3) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_4207_Inf__filter__not__bot,axiom,
    ! [A: $tType,B3: set(filter(A))] :
      ( ! [X7: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X7),B3))
         => ( pp(aa(set(filter(A)),bool,finite_finite2(filter(A)),X7))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X7) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_4208_folding__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite_folding_on(A,B,S2,F2)
    <=> ! [X3: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S2))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y4)),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,Y4)) ) ) ) ) ).

% folding_on_def
tff(fact_4209_folding__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,Y: A] :
      ( finite_folding_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),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,Y)) ) ) ) ) ).

% folding_on.comp_fun_commute_on
tff(fact_4210_folding__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),S2))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y2)),aa(A,fun(B,B),F2,X4)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X4)),aa(A,fun(B,B),F2,Y2)) ) ) )
     => finite_folding_on(A,B,S2,F2) ) ).

% folding_on.intro
tff(fact_4211_folding__on_Oempty,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),bot_bot(set(A))) = Z ) ) ).

% folding_on.empty
tff(fact_4212_folding__on_Oinfinite,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
       => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A3) = Z ) ) ) ).

% folding_on.infinite
tff(fact_4213_card__def,axiom,
    ! [B: $tType] : finite_card(B) = finite_folding_F(B,nat,aTP_Lamp_mb(B,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_4214_folding__on_Oinsert,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( 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),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),aa(set(A),B,finite_folding_F(A,B,F2,Z),A3)) ) ) ) ) ) ).

% folding_on.insert
tff(fact_4215_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),X: A,Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A3) = aa(B,B,aa(A,fun(B,B),F2,X),aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_4216_length__remdups__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),remdups(A,aa(list(list(A)),list(A),concat(A),Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))) ).

% length_remdups_concat
tff(fact_4217_folding__idem__on_Oinsert__idem,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( 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),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),aa(set(A),B,finite_folding_F(A,B,F2,Z),A3)) ) ) ) ) ).

% folding_idem_on.insert_idem
tff(fact_4218_Id__on__def,axiom,
    ! [A: $tType,A3: set(A)] : id_on(A,A3) = 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_mc(A,set(product_prod(A,A)))),A3)) ).

% Id_on_def
tff(fact_4219_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F2: fun(nat,set(A)),S2: set(A)] :
      ( ! [I2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F2,I2)),S2))
     => ( pp(aa(set(A),bool,finite_finite2(A),S2))
       => ( ? [N8: nat] :
              ( ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N8))
                 => ! [M5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N8))
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N2))
                       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F2,M5)),aa(nat,set(A),F2,N2))) ) ) )
              & ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
                 => ( aa(nat,set(A),F2,N8) = aa(nat,set(A),F2,N2) ) ) )
         => ( aa(nat,set(A),F2,aa(set(A),nat,finite_card(A),S2)) = 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_4220_top__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( top(C)
     => ! [X: D] : aa(D,C,top_top(fun(D,C)),X) = top_top(C) ) ).

% top_apply
tff(fact_4221_UNIV__I,axiom,
    ! [A: $tType,X: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),top_top(set(A)))) ).

% UNIV_I
tff(fact_4222_finite__Plus__UNIV__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( pp(aa(set(sum_sum(A,B)),bool,finite_finite2(sum_sum(A,B)),top_top(set(sum_sum(A,B)))))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
        & pp(aa(set(B),bool,finite_finite2(B),top_top(set(B)))) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_4223_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),top_top(set(option(A)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% finite_option_UNIV
tff(fact_4224_Int__UNIV,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = top_top(set(A)) )
    <=> ( ( A3 = top_top(set(A)) )
        & ( B3 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_4225_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),top_top(A)) = top_top(A) ) ).

% max_top2
tff(fact_4226_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),X) = top_top(A) ) ).

% max_top
tff(fact_4227_Pow__UNIV,axiom,
    ! [A: $tType] : pow(A,top_top(set(A))) = top_top(set(set(A))) ).

% Pow_UNIV
tff(fact_4228_remdups__eq__nil__iff,axiom,
    ! [A: $tType,X: list(A)] :
      ( ( remdups(A,X) = nil(A) )
    <=> ( X = nil(A) ) ) ).

% remdups_eq_nil_iff
tff(fact_4229_remdups__eq__nil__right__iff,axiom,
    ! [A: $tType,X: list(A)] :
      ( ( nil(A) = remdups(A,X) )
    <=> ( X = nil(A) ) ) ).

% remdups_eq_nil_right_iff
tff(fact_4230_set__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),remdups(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_remdups
tff(fact_4231_length__remdups__eq,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),remdups(A,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ( remdups(A,Xs) = Xs ) ) ).

% length_remdups_eq
tff(fact_4232_Id__onI,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2)),id_on(A,A3))) ) ).

% Id_onI
tff(fact_4233_distinct__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : distinct(A,remdups(A,Xs)) ).

% distinct_remdups
tff(fact_4234_remdups__id__iff__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( remdups(A,Xs) = Xs )
    <=> distinct(A,Xs) ) ).

% remdups_id_iff_distinct
tff(fact_4235_Collect__const,axiom,
    ! [A: $tType,P: bool] :
      ( ( pp(P)
       => ( aa(fun(A,bool),set(A),collect(A),aTP_Lamp_md(bool,fun(A,bool),P)) = top_top(set(A)) ) )
      & ( ~ pp(P)
       => ( aa(fun(A,bool),set(A),collect(A),aTP_Lamp_md(bool,fun(A,bool),P)) = bot_bot(set(A)) ) ) ) ).

% Collect_const
tff(fact_4236_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bu(fun(A,bool),fun(A,bool),P))))
      <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).

% finite_Collect_not
tff(fact_4237_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A3) = top_top(A) )
        <=> ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),top_top(A)))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Xa3)) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_4238_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_4239_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_4240_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_4241_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_4242_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_4243_Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_4244_surj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),N: 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)),N),F2)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_4245_finite__compl,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),uminus_uminus(set(A)),A3)))
      <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).

% finite_compl
tff(fact_4246_length__remdups__leq,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_remdups_leq
tff(fact_4247_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_4248_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)) = top_top(A) )
        <=> ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),top_top(A)))
             => ? [Xa3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),aa(B,A,F2,Xa3))) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_4249_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image2(B,A,aTP_Lamp_kv(A,fun(B,A),X)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ).

% range_constant
tff(fact_4250_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_4251_INT__constant,axiom,
    ! [B: $tType,A: $tType,A3: set(B),C2: set(A)] :
      ( ( ( A3 = bot_bot(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_lo(set(A),fun(B,set(A)),C2)),A3)) = top_top(set(A)) ) )
      & ( ( A3 != bot_bot(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_lo(set(A),fun(B,set(A)),C2)),A3)) = C2 ) ) ) ).

% INT_constant
tff(fact_4252_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_ord_lessThan(A,X)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_4253_INT__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C3: set(D),A3: set(C),B3: fun(D,set(C))] :
      ( ( ( C3 = bot_bot(set(D)) )
       => ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_lu(set(C),fun(fun(D,set(C)),fun(D,set(C))),A3),B3)),C3)) = top_top(set(C)) ) )
      & ( ( C3 != bot_bot(set(D)) )
       => ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_lu(set(C),fun(fun(D,set(C)),fun(D,set(C))),A3),B3)),C3)) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A3),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B3),C3))) ) ) ) ).

% INT_simps(2)
tff(fact_4254_INT__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C3: set(A),A3: fun(A,set(B)),B3: set(B)] :
      ( ( ( C3 = bot_bot(set(A)) )
       => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_lv(fun(A,set(B)),fun(set(B),fun(A,set(B))),A3),B3)),C3)) = top_top(set(B)) ) )
      & ( ( C3 != bot_bot(set(A)) )
       => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_lv(fun(A,set(B)),fun(set(B),fun(A,set(B))),A3),B3)),C3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),C3))),B3) ) ) ) ).

% INT_simps(1)
tff(fact_4255_INT__simps_I3_J,axiom,
    ! [E: $tType,F: $tType,C3: set(E),A3: fun(E,set(F)),B3: set(F)] :
      ( ( ( C3 = bot_bot(set(E)) )
       => ( aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),aa(set(F),fun(E,set(F)),aTP_Lamp_me(fun(E,set(F)),fun(set(F),fun(E,set(F))),A3),B3)),C3)) = top_top(set(F)) ) )
      & ( ( C3 != bot_bot(set(E)) )
       => ( aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),aa(set(F),fun(E,set(F)),aTP_Lamp_me(fun(E,set(F)),fun(set(F),fun(E,set(F))),A3),B3)),C3)) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),A3),C3))),B3) ) ) ) ).

% INT_simps(3)
tff(fact_4256_INT__simps_I4_J,axiom,
    ! [G2: $tType,H3: $tType,C3: set(H3),A3: set(G2),B3: fun(H3,set(G2))] :
      ( ( ( C3 = bot_bot(set(H3)) )
       => ( aa(set(set(G2)),set(G2),complete_Inf_Inf(set(G2)),aa(set(H3),set(set(G2)),image2(H3,set(G2),aa(fun(H3,set(G2)),fun(H3,set(G2)),aTP_Lamp_lx(set(G2),fun(fun(H3,set(G2)),fun(H3,set(G2))),A3),B3)),C3)) = top_top(set(G2)) ) )
      & ( ( C3 != bot_bot(set(H3)) )
       => ( aa(set(set(G2)),set(G2),complete_Inf_Inf(set(G2)),aa(set(H3),set(set(G2)),image2(H3,set(G2),aa(fun(H3,set(G2)),fun(H3,set(G2)),aTP_Lamp_lx(set(G2),fun(fun(H3,set(G2)),fun(H3,set(G2))),A3),B3)),C3)) = aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),A3),aa(set(set(G2)),set(G2),complete_Sup_Sup(set(G2)),aa(set(H3),set(set(G2)),image2(H3,set(G2),B3),C3))) ) ) ) ).

% INT_simps(4)
tff(fact_4257_range__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),X: B] :
      ( ( B2 = aa(B,A,F2,X) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))) ) ).

% range_eqI
tff(fact_4258_rangeI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: B] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X)),aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))) ).

% rangeI
tff(fact_4259_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),top_top(A))) ) ).

% top_greatest
tff(fact_4260_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A2))
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_4261_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A2))
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_4262_subset__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),top_top(set(A)))) ).

% subset_UNIV
tff(fact_4263_less__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F8: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less(filter(A)),F4),F8))
    <=> ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F8))
        & ~ pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F8),F4)) ) ) ).

% less_filter_def
tff(fact_4264_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_4265_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [X: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,X,Y) = top_top(set(A)) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_4266_insert__UNIV,axiom,
    ! [A: $tType,X: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_4267_distinct__remdups__id,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups(A,Xs) = Xs ) ) ).

% distinct_remdups_id
tff(fact_4268_infinite__UNIV__nat,axiom,
    ~ pp(aa(set(nat),bool,finite_finite2(nat),top_top(set(nat)))) ).

% infinite_UNIV_nat
tff(fact_4269_nat__not__finite,axiom,
    ~ pp(aa(set(nat),bool,finite_finite2(nat),top_top(set(nat)))) ).

% nat_not_finite
tff(fact_4270_Int__UNIV__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),top_top(set(A))) = A3 ).

% Int_UNIV_right
tff(fact_4271_Int__UNIV__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B3) = B3 ).

% Int_UNIV_left
tff(fact_4272_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_mf(A,bool)) ).

% UNIV_def
tff(fact_4273_UNIV__eq__I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X4: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
     => ( top_top(set(A)) = A3 ) ) ).

% UNIV_eq_I
tff(fact_4274_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X4: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),top_top(set(A)))) ).

% UNIV_witness
tff(fact_4275_remdups__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : remdups(A,remdups(A,Xs)) = remdups(A,Xs) ).

% remdups_remdups
tff(fact_4276_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A2)) ) ).

% top.extremum_strict
tff(fact_4277_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( ( A2 != top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),top_top(A))) ) ) ).

% top.not_eq_extremum
tff(fact_4278_infinite__UNIV__char__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ~ pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% infinite_UNIV_char_0
tff(fact_4279_ex__new__if__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
     => ( pp(aa(set(A),bool,finite_finite2(A),A3))
       => ? [A4: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3)) ) ) ).

% ex_new_if_finite
tff(fact_4280_finite__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% finite_UNIV
tff(fact_4281_Finite__Set_Ofinite__set,axiom,
    ! [A: $tType] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),top_top(set(set(A)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% Finite_Set.finite_set
tff(fact_4282_finite__Prod__UNIV,axiom,
    ! [B: $tType,A: $tType] :
      ( pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
     => ( pp(aa(set(B),bool,finite_finite2(B),top_top(set(B))))
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B))))) ) ) ).

% finite_Prod_UNIV
tff(fact_4283_finite__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B)))))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
        & pp(aa(set(B),bool,finite_finite2(B),top_top(set(B)))) ) ) ).

% finite_prod
tff(fact_4284_finite__fun__UNIVD2,axiom,
    ! [A: $tType,B: $tType] :
      ( pp(aa(set(fun(A,B)),bool,finite_finite2(fun(A,B)),top_top(set(fun(A,B)))))
     => pp(aa(set(B),bool,finite_finite2(B),top_top(set(B)))) ) ).

% finite_fun_UNIVD2
tff(fact_4285_remdups_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups(A,nil(A)) = nil(A) ).

% remdups.simps(1)
tff(fact_4286_Id__on__iff,axiom,
    ! [A: $tType,X: A,Y: A,A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),id_on(A,A3)))
    <=> ( ( X = Y )
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ) ).

% Id_on_iff
tff(fact_4287_Id__on__eqI,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( ( A2 = B2 )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),id_on(A,A3))) ) ) ).

% Id_on_eqI
tff(fact_4288_Id__onE,axiom,
    ! [A: $tType,C2: product_prod(A,A),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),C2),id_on(A,A3)))
     => ~ ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ) ) ).

% Id_onE
tff(fact_4289_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_mg(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_4290_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_mh(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_4291_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))))
     => ~ ! [X4: B] : B2 != aa(B,A,F2,X4) ) ).

% rangeE
tff(fact_4292_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_4293_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite_folding_on(A,list(A),top_top(set(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A))) ) ).

% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_4294_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( pp(aa(set(fun(A,B)),bool,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)) )
       => pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).

% finite_fun_UNIVD1
tff(fact_4295_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),B3: set(A),I: B] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))),B3))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,I)),B3)) ) ).

% range_subsetD
tff(fact_4296_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H))) ) ).

% not_UNIV_le_Icc
tff(fact_4297_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
     => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
       => ( A3 = top_top(set(A)) ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_4298_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_ord_atMost(A,H))) ) ).

% not_UNIV_le_Iic
tff(fact_4299_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_4300_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_4301_folding__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,Y: A] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
         => ( 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.comp_fun_idem_on
tff(fact_4302_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A3) ).

% Compl_eq_Diff_UNIV
tff(fact_4303_Inter__empty,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),bot_bot(set(set(A)))) = top_top(set(A)) ).

% Inter_empty
tff(fact_4304_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F2: fun(A,C)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,G),top_top(set(B)))))
     => pp(aa(set(C),bool,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_mi(fun(B,A),fun(fun(A,C),fun(B,C)),G),F2)),top_top(set(B))))) ) ).

% finite_range_imageI
tff(fact_4305_remove1__remdups,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( remove1(A,X,remdups(A,Xs)) = remdups(A,remove1(A,X,Xs)) ) ) ).

% remove1_remdups
tff(fact_4306_folding__idem__on_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
     => finite_folding_on(A,B,S2,F2) ) ).

% folding_idem_on.axioms(1)
tff(fact_4307_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: A,X: 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),A2),bot_bot(set(A))) )
     => ( aa(B,A,F2,X) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4308_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),C2: A] :
          ( ( ( A3 = bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kl(A,fun(B,A),C2)),A3)) = top_top(A) ) )
          & ( ( A3 != bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kl(A,fun(B,A),C2)),A3)) = C2 ) ) ) ) ).

% INF_constant
tff(fact_4309_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_4310_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),uminus_uminus(set(B)),A3)))) ) ).

% surj_Compl_image_subset
tff(fact_4311_length__remdups__card__conv,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),remdups(A,Xs)) = aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) ).

% length_remdups_card_conv
tff(fact_4312_finite__range__Some,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% finite_range_Some
tff(fact_4313_notin__range__Some,axiom,
    ! [A: $tType,X: option(A)] :
      ( ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),X),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))))
    <=> ( X = none(A) ) ) ).

% notin_range_Some
tff(fact_4314_INT__empty,axiom,
    ! [B: $tType,A: $tType,B3: 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),B3),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_4315_UN__UN__finite__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aTP_Lamp_mj(fun(nat,set(A)),fun(nat,set(A)),A3)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_4316_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_4317_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ).

% finite_UNIV_card_ge_0
tff(fact_4318_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_4319_UN__finite__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),C3: set(A)] :
      ( ! [N2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),C3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),top_top(set(nat))))),C3)) ) ).

% UN_finite_subset
tff(fact_4320_UN__finite2__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B3: 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),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),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),A3),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),B3),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_4321_INT__extend__simps_I3_J,axiom,
    ! [F: $tType,E: $tType,C3: set(E),A3: fun(E,set(F)),B3: set(F)] :
      ( ( ( C3 = bot_bot(set(E)) )
       => ( aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),A3),C3))),B3) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),top_top(set(F))),B3) ) )
      & ( ( C3 != bot_bot(set(E)) )
       => ( aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),A3),C3))),B3) = aa(set(set(F)),set(F),complete_Inf_Inf(set(F)),aa(set(E),set(set(F)),image2(E,set(F),aa(set(F),fun(E,set(F)),aTP_Lamp_me(fun(E,set(F)),fun(set(F),fun(E,set(F))),A3),B3)),C3)) ) ) ) ).

% INT_extend_simps(3)
tff(fact_4322_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))))) ) ).

% card_range_greater_zero
tff(fact_4323_UN__finite2__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B3: fun(nat,set(A)),K: nat] :
      ( ! [N2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))))))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),top_top(set(nat)))))) ) ).

% UN_finite2_subset
tff(fact_4324_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_mk(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_4325_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_4326_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_4327_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_ml(fun(A,A),fun(nat,A),F2)),top_top(set(nat)))) ) ).

% cclfp_def
tff(fact_4328_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_4329_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_4330_range__mult,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero(real) )
       => ( aa(set(real),set(real),image2(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = aa(set(real),set(real),aa(real,fun(set(real),set(real)),insert2(real),zero_zero(real)),bot_bot(set(real))) ) )
      & ( ( A2 != zero_zero(real) )
       => ( aa(set(real),set(real),image2(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = top_top(set(real)) ) ) ) ).

% range_mult
tff(fact_4331_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: bool] :
      ( ( pp(P)
       => ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_mm(bool,fun(A,fun(B,bool)),P))) = top_top(set(product_prod(A,B))) ) )
      & ( ~ pp(P)
       => ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_mm(bool,fun(A,fun(B,bool)),P))) = bot_bot(set(product_prod(A,B))) ) ) ) ).

% Collect_const_case_prod
tff(fact_4332_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,bool),set(A),collect(A),top_top(fun(A,bool))) ).

% top_set_def
tff(fact_4333_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),top_top(set(product_prod(A,B))))) ) ).

% top_empty_eq2
tff(fact_4334_infinite__UNIV__int,axiom,
    ~ pp(aa(set(int),bool,finite_finite2(int),top_top(set(int)))) ).

% infinite_UNIV_int
tff(fact_4335_infinite__UNIV__listI,axiom,
    ! [A: $tType] : ~ pp(aa(set(list(A)),bool,finite_finite2(list(A)),top_top(set(list(A))))) ).

% infinite_UNIV_listI
tff(fact_4336_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_4337_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(real,real,root(N),X) = zero_zero(real) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(real,real,root(N),X) = the_inv_into(real,real,top_top(set(real)),aTP_Lamp_mn(nat,fun(real,real),N),X) ) ) ) ).

% root_def
tff(fact_4338_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [X: A] : top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ).

% perfect_space_class.UNIV_not_singleton
tff(fact_4339_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),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_mo(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F2),R))) ).

% mlex_eq
tff(fact_4340_these__insert__Some,axiom,
    ! [A: $tType,X: A,A3: 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),X)),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),these(A,A3)) ).

% these_insert_Some
tff(fact_4341_top1I,axiom,
    ! [A: $tType,X: A] : pp(aa(A,bool,top_top(fun(A,bool)),X)) ).

% top1I
tff(fact_4342_top2I,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X),Y)) ).

% top2I
tff(fact_4343_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_4344_these__image__Some__eq,axiom,
    ! [A: $tType,A3: set(A)] : these(A,aa(set(A),set(option(A)),image2(A,option(A),some(A)),A3)) = A3 ).

% these_image_Some_eq
tff(fact_4345_these__insert__None,axiom,
    ! [A: $tType,A3: 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)),none(A)),A3)) = these(A,A3) ).

% these_insert_None
tff(fact_4346_in__these__eq,axiom,
    ! [A: $tType,X: A,A3: set(option(A))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),these(A,A3)))
    <=> pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),aa(A,option(A),some(A),X)),A3)) ) ).

% in_these_eq
tff(fact_4347_Option_Othese__def,axiom,
    ! [A: $tType,A3: set(option(A))] : these(A,A3) = aa(set(option(A)),set(A),image2(option(A),A,the2(A)),aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_mp(set(option(A)),fun(option(A),bool),A3))) ).

% Option.these_def
tff(fact_4348_these__empty__eq,axiom,
    ! [A: $tType,B3: set(option(A))] :
      ( ( these(A,B3) = bot_bot(set(A)) )
    <=> ( ( B3 = bot_bot(set(option(A))) )
        | ( B3 = 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_4349_these__not__empty__eq,axiom,
    ! [A: $tType,B3: set(option(A))] :
      ( ( these(A,B3) != bot_bot(set(A)) )
    <=> ( ( B3 != bot_bot(set(option(A))) )
        & ( B3 != 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_4350_Some__image__these__eq,axiom,
    ! [A: $tType,A3: set(option(A))] : aa(set(A),set(option(A)),image2(A,option(A),some(A)),these(A,A3)) = aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_mp(set(option(A)),fun(option(A),bool),A3)) ).

% Some_image_these_eq
tff(fact_4351_mlex__leq,axiom,
    ! [A: $tType,F2: fun(A,nat),X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F2,R))) ) ) ).

% mlex_leq
tff(fact_4352_mlex__less,axiom,
    ! [A: $tType,F2: fun(A,nat),X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F2,R))) ) ).

% mlex_less
tff(fact_4353_mlex__iff,axiom,
    ! [A: $tType,X: A,Y: A,F2: fun(A,nat),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F2,R)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y)))
        | ( ( aa(A,nat,F2,X) = aa(A,nat,F2,Y) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)) ) ) ) ).

% mlex_iff
tff(fact_4354_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
         => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_4355_at__within__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] : topolo174197925503356063within(A,A2,bot_bot(set(A))) = bot_bot(filter(A)) ) ).

% at_within_empty
tff(fact_4356_at__discrete,axiom,
    ! [A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [X: A,S2: set(A)] : topolo174197925503356063within(A,X,S2) = bot_bot(filter(A)) ) ).

% at_discrete
tff(fact_4357_DERIV__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F9: A,X: A,S: set(A),T2: set(A)] :
          ( has_field_derivative(A,F2,F9,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => has_field_derivative(A,F2,F9,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% DERIV_subset
tff(fact_4358_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,X: A,S: set(A),T2: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => has_field_derivative(A,F2,Y,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% has_field_derivative_subset
tff(fact_4359_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F4: filter(A)] : has_field_derivative(A,aTP_Lamp_mq(A,fun(A,A),K),zero_zero(A),F4) ) ).

% DERIV_const
tff(fact_4360_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H4)),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H4)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_4361_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H4)),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H4))),aa(real,real,F2,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_4362_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))),aa(real,real,F2,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_4363_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_4364_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),zero_zero(real))) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_4365_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3)) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,A2)),aa(real,real,F2,B2))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_4366_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_4367_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))),aa(real,real,F2,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_4368_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H4))),aa(real,real,F2,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_4369_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),X),H4)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_4370_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_mr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D6),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,X)),aa(A,A,G,X))),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_4371_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_ms(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,F2,X))),D6)),aa(A,A,inverse_inverse(A),aa(A,A,F2,X)))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_4372_at__neq__bot,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [A2: A] : topolo174197925503356063within(A,A2,top_top(set(A))) != bot_bot(filter(A)) ) ).

% at_neq_bot
tff(fact_4373_at__le,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T2: set(A),X: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),T2))
         => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),topolo174197925503356063within(A,X,S)),topolo174197925503356063within(A,X,T2))) ) ) ).

% at_le
tff(fact_4374_trivial__limit__at__left__real,axiom,
    ! [A: $tType] :
      ( ( dense_order(A)
        & no_bot(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : topolo174197925503356063within(A,X,set_ord_lessThan(A,X)) != bot_bot(filter(A)) ) ).

% trivial_limit_at_left_real
tff(fact_4375_MVT2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F9: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2))
             => has_field_derivative(real,F2,aa(real,real,F9,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
       => ? [Z3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),B2))
            & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(real,real,F9,Z3)) ) ) ) ) ).

% MVT2
tff(fact_4376_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ( ! [Y2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y2))),D2))
             => ( aa(real,real,F2,X) = aa(real,real,F2,Y2) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_4377_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_4378_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,S: set(A)] :
          ( ( X != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_inverse
tff(fact_4379_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A,S: set(A),N: nat] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_mt(fun(A,A),fun(nat,fun(A,A)),F2),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),times_times(A),D6),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_power
tff(fact_4380_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ( ! [Y2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y2))),D2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F2,Y2)),aa(real,real,F2,X))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_4381_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ( ! [Y2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y2))),D2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F2,X)),aa(real,real,F2,Y2))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_4382_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,ln_ln(real),divide_divide(real,one_one(real),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_4383_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_mu(nat,fun(real,real),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,X,S)) ).

% DERIV_pow
tff(fact_4384_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
           => ( topolo174197925503356063within(A,X,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,X,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_4385_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( topolo174197925503356063within(A,B2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_4386_trivial__limit__at__left__bot,axiom,
    ! [A: $tType] :
      ( ( order_bot(A)
        & topolo1944317154257567458pology(A) )
     => ( topolo174197925503356063within(A,bot_bot(A),set_ord_lessThan(A,bot_bot(A))) = bot_bot(filter(A)) ) ) ).

% trivial_limit_at_left_bot
tff(fact_4387_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S: set(A),G: fun(A,A),E3: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E3,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_mr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),E3),aa(A,A,F2,X))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_4388_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_ms(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_4389_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K4: real,C2: fun(nat,A),F2: fun(A,A),F9: A,Z: A] :
          ( ! [Z3: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K4))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_gi(fun(nat,A),fun(A,fun(nat,A)),C2),Z3),aa(A,A,F2,Z3)) )
         => ( has_field_derivative(A,F2,F9,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K4))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_mv(fun(nat,A),fun(A,fun(nat,A)),C2),Z),F9) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_4390_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Z))
     => has_field_derivative(real,aTP_Lamp_mw(real,fun(real,real),R2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,Z,aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_4391_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K4: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gi(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_mv(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_mx(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K4)))
               => has_field_derivative(A,aTP_Lamp_my(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_mv(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_4392_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K4: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gi(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K4)))
           => has_field_derivative(A,aTP_Lamp_my(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_mv(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_4393_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K4: real,C2: fun(nat,A),Z: A] :
          ( ! [Z3: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K4))
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_gi(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K4))
           => has_field_derivative(A,aTP_Lamp_my(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_mv(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_4394_DERIV__log,axiom,
    ! [X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,log(B2),divide_divide(real,one_one(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_4395_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,G,X)))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_mz(fun(real,real),fun(real,fun(real,real)),G),R2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,aa(real,real,G,X),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_4396_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,F2: fun(real,real),R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,G,X)))
       => ( has_field_derivative(real,F2,R2,topolo174197925503356063within(real,X,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_na(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,X),aa(real,real,F2,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),aa(real,real,ln_ln(real),aa(real,real,G,X)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,F2,X)),aa(real,real,G,X)))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_4397_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_4398_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => has_field_derivative(real,arcosh(real),divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,X,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_4399_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,sqrt,divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_4400_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_4401_has__field__derivative__tanh,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [G: fun(A9,A9),X: A9,Db: A9,S: set(A9)] :
          ( ( cosh(A9,aa(A9,A9,G,X)) != zero_zero(A9) )
         => ( has_field_derivative(A9,G,Db,topolo174197925503356063within(A9,X,S))
           => has_field_derivative(A9,aTP_Lamp_nb(fun(A9,A9),fun(A9,A9),G),aa(A9,A9,aa(A9,fun(A9,A9),times_times(A9),aa(A9,A9,aa(A9,fun(A9,A9),minus_minus(A9),one_one(A9)),aa(nat,A9,aa(A9,fun(nat,A9),power_power(A9),aa(A9,A9,tanh(A9),aa(A9,A9,G,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Db),topolo174197925503356063within(A9,X,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_4402_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D6: real] :
      ( ( X != zero_zero(real) )
     => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( D6 = divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
       => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
           => ( D6 = divide_divide(real,aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
         => has_field_derivative(real,sqrt,D6,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_4403_artanh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => has_field_derivative(real,artanh(real),divide_divide(real,one_one(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(real,X,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_4404_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_4405_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_4406_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,N: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M5: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),X4),topolo174197925503356063within(real,X4,top_top(set(real)))) )
     => ? [T7: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),X)))
          & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_nc(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_4407_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,N: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M5: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),X4),topolo174197925503356063within(real,X4,top_top(set(real))))
       => ? [T7: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_nc(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_4408_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( ( X != zero_zero(real) )
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_4409_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H),zero_zero(real)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M5: nat,T7: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),H),T7))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),zero_zero(real))) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
           => ? [T7: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H),T7))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),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_nd(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_4410_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M5: nat,T7: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T7))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),H)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
         => ? [T7: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T7))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),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_nd(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_4411_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M5: nat,T7: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T7))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),H)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
           => ? [T7: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T7))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),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_nd(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_4412_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( X != zero_zero(real) )
         => ( ! [M5: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),X4),topolo174197925503356063within(real,X4,top_top(set(real))))
           => ? [T7: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T7)))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),X)))
                & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_nc(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_4413_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M5: nat,T7: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),X))) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
       => ? [T7: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_nc(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_4414_Taylor__down,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M5: nat,T7: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),T7))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),B2))
             => ? [T7: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),T7))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),C2))
                  & ( aa(real,real,F2,A2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_ne(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_4415_Taylor__up,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M5: nat,T7: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),T7))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),B2))
             => ? [T7: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),T7))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),B2))
                  & ( aa(real,real,F2,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_ne(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_4416_Taylor,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M5: nat,T7: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),T7))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),B2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),B2))
                 => ( ( X != C2 )
                   => ? [T7: real] :
                        ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),C2))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),T7))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),C2)) ) )
                        & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),C2))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),T7))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T7),X)) ) )
                        & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_nf(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),X)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T7),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),C2)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_4417_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B3: real] :
      ( ! [M5: nat,T7: real] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T7))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),H)) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
     => ( ( N = aa(nat,nat,suc,K) )
       => ! [M2: nat,T8: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T8))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T8),H)) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_nh(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),N),Diff),B3),M2),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T8)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_ni(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T8)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M2))))),aa(real,real,aa(real,fun(real,real),times_times(real),B3),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),T8),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M2))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M2))))))),topolo174197925503356063within(real,T8,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_4418_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => has_field_derivative(real,aTP_Lamp_nk(real,real),suminf(real,aTP_Lamp_nl(real,fun(nat,real),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_4419_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D6: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( X != zero_zero(real) )
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
             => ( D6 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
               => ( D6 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
               => ( D6 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(N),D6,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_4420_in__finite__psubset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A3),B3)),finite_psubset(A)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
        & pp(aa(set(A),bool,finite_finite2(A),B3)) ) ) ).

% in_finite_psubset
tff(fact_4421_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_4422_DERIV__power__series_H,axiom,
    ! [R: real,F2: fun(nat,real),X0: real] :
      ( ! [X4: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R)))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_nm(fun(nat,real),fun(real,fun(nat,real)),F2),X4)) )
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
         => has_field_derivative(real,aTP_Lamp_no(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_nm(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_4423_in__measure,axiom,
    ! [A: $tType,X: A,Y: A,F2: fun(A,nat)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measure(A,F2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y))) ) ).

% in_measure
tff(fact_4424_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or5935395276787703475ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),U)) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_4425_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_4426_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_4427_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A2,B2) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_4428_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or5935395276787703475ssThan(A,A2,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% infinite_Ioo_iff
tff(fact_4429_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_4430_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_4431_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_4432_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_4433_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or5935395276787703475ssThan(A,A2,B2))) ) ) ).

% infinite_Ioo
tff(fact_4434_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_4435_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_4436_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_4437_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_4438_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)),set_ord_atMost(A,L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(1)
tff(fact_4439_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_4440_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_4441_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_4442_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,Y)),aa(real,real,arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_4443_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,X)),aa(real,real,arccos,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X)) ) ) ) ).

% arccos_less_mono
tff(fact_4444_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,X: real,Y: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),set_or5935395276787703475ssThan(real,A2,B2)))
       => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y),set_or5935395276787703475ssThan(real,A2,B2)))
         => ( ! [X4: real] :
                ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),set_or5935395276787703475ssThan(real,A2,B2)))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) )
           => ( aa(real,real,F2,X) = aa(real,real,F2,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_4445_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,arccos,Y)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_4446_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => ( sin(real,aa(real,real,arccos,X)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_4447_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_np(set(A),fun(set(A),bool)))) ).

% finite_psubset_def
tff(fact_4448_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G6: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,G,X)),one_one(real)))
           => ( has_derivative(A,real,G,G6,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_nq(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_nr(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G6),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_4449_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G6: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,G,X)),one_one(real)))
           => ( has_derivative(A,real,G,G6,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_ns(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_nt(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G6),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_4450_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G6: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X)))
         => ( has_derivative(A,real,G,G6,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_nu(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_nv(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G6),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_4451_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K4: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_mx(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K4)))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_nx(fun(nat,A),fun(A,fun(A,A)),C2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_4452_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_or5935395276787703475ssThan(nat,L,U))) ).

% finite_greaterThanLessThan
tff(fact_4453_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or5935395276787703475ssThan(int,L,U))) ).

% finite_greaterThanLessThan_int
tff(fact_4454_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ny(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_4455_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_nz(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_4456_power__tendsto__0__iff,axiom,
    ! [A: $tType,N: nat,F2: fun(A,real),F4: filter(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_oa(nat,fun(fun(A,real),fun(A,real)),N),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_4457_LIM__not__zero,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topolo8386298272705272623_space(A)
        & zero(Aa)
        & topological_t2_space(Aa) )
     => ! [K: Aa,A2: A] :
          ( ( K != zero_zero(Aa) )
         => ~ filterlim(A,Aa,aTP_Lamp_ob(Aa,fun(A,Aa),K),topolo7230453075368039082e_nhds(Aa,zero_zero(Aa)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_not_zero
tff(fact_4458_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,C),B2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B2),F4)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_oc(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),topolo7230453075368039082e_nhds(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),F4) ) ) ) ).

% tendsto_Pair
tff(fact_4459_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F4: filter(A)] : has_derivative(A,B,aTP_Lamp_od(B,fun(A,B),C2),aTP_Lamp_oe(A,B),F4) ) ).

% has_derivative_const
tff(fact_4460_Lim__transform__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),G: fun(B,A),F4: filter(B),A2: A] :
          ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_of(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
          <=> filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A2),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_4461_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_og(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_4462_Lim__transform2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A2: A,F4: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_of(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A2),F4) ) ) ) ).

% Lim_transform2
tff(fact_4463_Lim__transform,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(B,A),A2: A,F4: filter(B),F2: fun(B,A)] :
          ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oh(fun(B,A),fun(fun(B,A),fun(B,A)),G),F2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F4) ) ) ) ).

% Lim_transform
tff(fact_4464_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_og(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_4465_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_og(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_4466_tendsto__add__zero,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(D,B),F4: filter(D),G: fun(D,B)] :
          ( filterlim(D,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_oi(fun(D,B),fun(fun(D,B),fun(D,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_4467_tendsto__mult__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D,A),F4: filter(D),G: fun(D,A)] :
          ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => ( filterlim(D,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_oj(fun(D,A),fun(fun(D,A),fun(D,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_4468_tendsto__mult__left__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D,A),F4: filter(D),C2: A] :
          ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => filterlim(D,A,aa(A,fun(D,A),aTP_Lamp_ok(fun(D,A),fun(A,fun(D,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_4469_tendsto__mult__right__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(D,A),F4: filter(D),C2: A] :
          ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => filterlim(D,A,aa(A,fun(D,A),aTP_Lamp_ol(fun(D,A),fun(A,fun(D,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_4470_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_om(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_4471_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_om(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_iff
tff(fact_4472_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,real,aTP_Lamp_om(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_norm_zero
tff(fact_4473_tendsto__sgn,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),L: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
         => ( ( L != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_on(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,sgn_sgn(A),L)),F4) ) ) ) ).

% tendsto_sgn
tff(fact_4474_tendsto__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A2: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( A2 != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_oo(fun(B,A),fun(B,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,inverse_inverse(A),A2)),F4) ) ) ) ).

% tendsto_inverse
tff(fact_4475_tendsto__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),A2: A,F4: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F4)
           => ( ( B2 != zero_zero(A) )
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_op(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),topolo7230453075368039082e_nhds(A,divide_divide(A,A2,B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_4476_tendsto__divide__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),F4: filter(B),C2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_oq(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_4477_tendsto__null__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I6: set(B),F2: fun(A,fun(B,C)),F4: filter(A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I6))
             => filterlim(A,C,aa(B,fun(A,C),aTP_Lamp_or(fun(A,fun(B,C)),fun(B,fun(A,C)),F2),I2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) )
         => filterlim(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_os(set(B),fun(fun(A,fun(B,C)),fun(A,C)),I6),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ).

% tendsto_null_sum
tff(fact_4478_nhds__neq__bot,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] : topolo7230453075368039082e_nhds(A,A2) != bot_bot(filter(A)) ) ).

% nhds_neq_bot
tff(fact_4479_tendsto__bot,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),A2: A] : filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),bot_bot(filter(B))) ) ).

% tendsto_bot
tff(fact_4480_tendsto__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(B),F2: fun(B,A),A2: A,B2: A] :
          ( ( F4 != bot_bot(filter(B)) )
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
           => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,B2),F4)
             => ( A2 = B2 ) ) ) ) ) ).

% tendsto_unique
tff(fact_4481_tendsto__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(B),A2: A,B2: A] :
          ( ( F4 != bot_bot(filter(B)) )
         => ( filterlim(B,A,aTP_Lamp_ot(A,fun(B,A),A2),topolo7230453075368039082e_nhds(A,B2),F4)
          <=> ( A2 = B2 ) ) ) ) ).

% tendsto_const_iff
tff(fact_4482_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( cos(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_ou(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F4) ) ) ) ).

% tendsto_tan
tff(fact_4483_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_ov(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F4) ) ) ) ).

% tendsto_cot
tff(fact_4484_tendsto__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),A2: A,F4: filter(C)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( cosh(A,A2) != zero_zero(A) )
           => filterlim(C,A,aTP_Lamp_ow(fun(C,A),fun(C,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tanh(A),A2)),F4) ) ) ) ).

% tendsto_tanh
tff(fact_4485_tendsto__arcosh,axiom,
    ! [B: $tType,F2: fun(B,real),A2: real,F4: filter(B)] :
      ( filterlim(B,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
       => filterlim(B,real,aTP_Lamp_ox(fun(B,real),fun(B,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ).

% tendsto_arcosh
tff(fact_4486_has__derivative__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),X: A,S: set(A),T2: set(A)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% has_derivative_subset
tff(fact_4487_tendsto__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),L: filter(B),X: A,S2: set(A),T3: set(A)] :
          ( filterlim(A,B,F2,L,topolo174197925503356063within(A,X,S2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),S2))
           => filterlim(A,B,F2,L,topolo174197925503356063within(A,X,T3)) ) ) ) ).

% tendsto_within_subset
tff(fact_4488_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_oy(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_4489_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_oy(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_4490_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_oy(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_4491_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F4: filter(A),N: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_oz(fun(A,B),fun(nat,fun(A,B)),F2),N),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_4492_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D6),topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_pa(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_4493_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
         => ( ( A2 != one_one(real) )
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pb(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_4494_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),one_one(real)))
         => filterlim(A,real,aTP_Lamp_pc(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_4495_LIM__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A2: A,F2: fun(A,D),L5: D] :
          ( nO_MATCH(C,A,zero_zero(C),A2)
         => ( filterlim(A,D,F2,topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
          <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_pd(A,fun(fun(A,D),fun(A,D)),A2),F2),topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_4496_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => ? [S4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S4))
                & ! [X2: A] :
                    ( ( ( X2 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A2))),S4)) )
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X2)),L5))),R2)) ) ) ) ) ) ).

% LIM_D
tff(fact_4497_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [S8: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S8))
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A2))),S8)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X4)),L5))),R3)) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_4498_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [S7: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S7))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A2))),S7)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X3)),L5))),R5)) ) ) ) ) ) ).

% LIM_eq
tff(fact_4499_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A2: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
         => ( ! [X4: A] :
                ( ( X4 != A2 )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A2))),R))
                 => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_4500_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A2: A,D6: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_pe(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_pf(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_4501_LIM__fun__gt__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [R3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
            & ! [X2: real] :
                ( ( ( X2 != C2 )
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X2))),R3)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,F2,X2))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_4502_LIM__fun__not__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
            & ! [X2: real] :
                ( ( ( X2 != C2 )
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X2))),R3)) )
               => ( aa(real,real,F2,X2) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_4503_LIM__fun__less__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [R3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
            & ! [X2: real] :
                ( ( ( X2 != C2 )
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X2))),R3)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,X2)),zero_zero(real))) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_4504_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D4))
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A2))),D4)) )
                     => ( aa(A,B,F2,X4) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_pg(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_4505_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F4: fun(A,B),X: A] :
          ( has_derivative(A,B,aTP_Lamp_oe(A,B),F4,topolo174197925503356063within(A,X,top_top(set(A))))
         => ! [X2: A] : aa(A,B,F4,X2) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_4506_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [T2: set(A),G: fun(A,B),G6: fun(A,fun(A,B)),F2: fun(C,A),S: set(C),X: C,F9: fun(C,A)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),T2))
             => has_derivative(A,B,G,aa(A,fun(A,B),G6,X4),topolo174197925503356063within(A,X4,T2)) )
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F2),S)),T2))
           => ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X),S))
             => ( has_derivative(C,A,F2,F9,topolo174197925503356063within(C,X,S))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ph(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_pi(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G6),F2),X),F9),topolo174197925503356063within(C,X,S)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_4507_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_pa(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_4508_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D6: A,X: A] :
          ( has_field_derivative(A,F2,D6,topolo174197925503356063within(A,X,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_pa(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D6),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_4509_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_pj(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_4510_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
         => ( ! [H5: A] :
                ( ( H5 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H5)),K))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,H5))),aa(real,real,aa(real,fun(real,real),times_times(real),K4),real_V7770717601297561774m_norm(A,H5)))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_4511_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),A2: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_pk(fun(A,B),fun(A,fun(A,B)),F2),A2),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_4512_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,I))))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_4513_has__derivative__divide_H,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),F9: fun(C,A),X: C,S2: set(C),G: fun(C,A),G6: fun(C,A)] :
          ( has_derivative(C,A,F2,F9,topolo174197925503356063within(C,X,S2))
         => ( has_derivative(C,A,G,G6,topolo174197925503356063within(C,X,S2))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_pl(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_pm(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F2),F9),X),G),G6),topolo174197925503356063within(C,X,S2)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_4514_has__derivative__inverse,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(C,A),X: C,F9: fun(C,A),S2: set(C)] :
          ( ( aa(C,A,F2,X) != zero_zero(A) )
         => ( has_derivative(C,A,F2,F9,topolo174197925503356063within(C,X,S2))
           => has_derivative(C,A,aTP_Lamp_pn(fun(C,A),fun(C,A),F2),aa(fun(C,A),fun(C,A),aa(C,fun(fun(C,A),fun(C,A)),aTP_Lamp_po(fun(C,A),fun(C,fun(fun(C,A),fun(C,A))),F2),X),F9),topolo174197925503356063within(C,X,S2)) ) ) ) ).

% has_derivative_inverse
tff(fact_4515_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,S2: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_pp(A,fun(A,A),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_inverse'
tff(fact_4516_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S))
         => ( ! [X4: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X4)),S))
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_gi(fun(nat,A),fun(A,fun(nat,A)),A2),X4),aa(A,A,F2,X4)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_4517_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S))
         => ( ! [X4: A] :
                ( ( X4 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X4)),S))
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_gi(fun(nat,A),fun(A,fun(nat,A)),A2),X4),aa(A,A,F2,X4)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_4518_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F2: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
         => ( summable(real,F2)
           => ( ! [H5: A,N2: nat] :
                  ( ( H5 != zero_zero(A) )
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H5)),K))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H5),N2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N2)),real_V7770717601297561774m_norm(A,H5)))) ) )
             => filterlim(A,B,aTP_Lamp_pq(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_4519_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G6: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X)))
         => ( has_derivative(A,real,G,G6,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_pr(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_ps(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G6),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_ln
tff(fact_4520_has__derivative__divide,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(C,A),F9: fun(C,A),X: C,S2: set(C),G: fun(C,A),G6: fun(C,A)] :
          ( has_derivative(C,A,F2,F9,topolo174197925503356063within(C,X,S2))
         => ( has_derivative(C,A,G,G6,topolo174197925503356063within(C,X,S2))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_pt(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_pu(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F2),F9),X),G),G6),topolo174197925503356063within(C,X,S2)) ) ) ) ) ).

% has_derivative_divide
tff(fact_4521_has__derivative__prod,axiom,
    ! [B: $tType,I7: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [I6: set(I7),F2: fun(I7,fun(A,B)),F9: fun(I7,fun(A,B)),X: A,S2: set(A)] :
          ( ! [I2: I7] :
              ( pp(aa(set(I7),bool,aa(I7,fun(set(I7),bool),member(I7),I2),I6))
             => has_derivative(A,B,aa(I7,fun(A,B),F2,I2),aa(I7,fun(A,B),F9,I2),topolo174197925503356063within(A,X,S2)) )
         => has_derivative(A,B,aa(fun(I7,fun(A,B)),fun(A,B),aTP_Lamp_pw(set(I7),fun(fun(I7,fun(A,B)),fun(A,B)),I6),F2),aa(A,fun(A,B),aa(fun(I7,fun(A,B)),fun(A,fun(A,B)),aa(fun(I7,fun(A,B)),fun(fun(I7,fun(A,B)),fun(A,fun(A,B))),aTP_Lamp_py(set(I7),fun(fun(I7,fun(A,B)),fun(fun(I7,fun(A,B)),fun(A,fun(A,B)))),I6),F2),F9),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_prod
tff(fact_4522_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G6: fun(A,real),X: A,X6: set(A),F2: fun(A,real),F9: fun(A,real)] :
          ( has_derivative(A,real,G,G6,topolo174197925503356063within(A,X,X6))
         => ( has_derivative(A,real,F2,F9,topolo174197925503356063within(A,X,X6))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X)))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pz(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_qa(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G6),X),F2),F9),topolo174197925503356063within(A,X,X6)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_4523_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,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_qb(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F9),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_4524_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_qb(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F9),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_within
tff(fact_4525_summable__Leibniz_I2_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(nat,real,A2,zero_zero(nat))))
         => ! [N5: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_qc(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qc(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qc(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5)),one_one(nat))))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_4526_summable__Leibniz_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,A2,zero_zero(nat))),zero_zero(real)))
         => ! [N5: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_qc(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qc(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5)),one_one(nat)))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qc(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_4527_trivial__limit__sequentially,axiom,
    at_top(nat) != bot_bot(filter(nat)) ).

% trivial_limit_sequentially
tff(fact_4528_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qd(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_4529_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qe(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_4530_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qf(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_4531_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_oe(A,B)) ) ).

% bounded_linear_zero
tff(fact_4532_trivial__limit__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_top(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_top_linorder
tff(fact_4533_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: fun(nat,A),X: A] :
          ( topological_monoseq(A,A2)
         => ( filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,N5)),X))
                & ! [M2: nat,N5: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,M2)),aa(nat,A,A2,N5))) ) )
              | ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,A2,N5)))
                & ! [M2: nat,N5: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,N5)),aa(nat,A,A2,M2))) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_4534_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ? [U3: fun(nat,A)] :
              ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,U3,N5)),X))
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_4535_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ? [U3: fun(nat,A)] :
              ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,U3,N5)))
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_4536_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N4: nat,X6: fun(nat,A),Y5: fun(nat,A),X: A,Y: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,Y5,N2))) )
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( filterlim(nat,A,Y5,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ).

% lim_mono
tff(fact_4537_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,Y5: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( filterlim(nat,A,Y5,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N8: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,Y5,N2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ).

% LIMSEQ_le
tff(fact_4538_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,M4: nat,C3: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N2)),C3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),C3)) ) ) ) ).

% Lim_bounded
tff(fact_4539_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,N4: nat,C3: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(nat,A,F2,N2))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),L)) ) ) ) ).

% Lim_bounded2
tff(fact_4540_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(nat,A,X6,N2))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X)) ) ) ) ).

% LIMSEQ_le_const
tff(fact_4541_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),A2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2)) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_4542_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N2: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,B2,N2)),S))
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(set(A),A,complete_Sup_Sup(A),S))) ) ) ) ).

% Sup_lim
tff(fact_4543_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N2: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,B2,N2)),S))
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S)),A2)) ) ) ) ).

% Inf_lim
tff(fact_4544_Inf__as__limit,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ? [U3: fun(nat,A)] :
              ( ! [N5: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,U3,N5)),A3))
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,aa(set(A),A,complete_Inf_Inf(A),A3)),at_top(nat)) ) ) ) ).

% Inf_as_limit
tff(fact_4545_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_4546_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N3)),aa(nat,A,X6,aa(nat,nat,suc,N3))))
            | ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N3))),aa(nat,A,X6,N3))) ) ) ) ).

% monoseq_Suc
tff(fact_4547_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N2))),aa(nat,A,X6,N2)))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI2
tff(fact_4548_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,X6,aa(nat,nat,suc,N2))))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI1
tff(fact_4549_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_qg(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_4550_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [M3: nat,N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,M3)),aa(nat,A,X6,N3))) )
            | ! [M3: nat,N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N3)),aa(nat,A,X6,M3))) ) ) ) ) ).

% monoseq_def
tff(fact_4551_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M5: nat,N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,X6,M5))) )
         => topological_monoseq(A,X6) ) ) ).

% monoI2
tff(fact_4552_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M5: nat,N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,M5)),aa(nat,A,X6,N2))) )
         => topological_monoseq(A,X6) ) ) ).

% monoI1
tff(fact_4553_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C2))
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_4554_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C2))
     => filterlim(nat,nat,aTP_Lamp_qh(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_4555_has__derivative__within__singleton__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,G,topolo174197925503356063within(A,X,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
        <=> real_V3181309239436604168linear(A,B,G) ) ) ).

% has_derivative_within_singleton_iff
tff(fact_4556_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_qi(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_4557_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_qj(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_4558_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X6: fun(nat,A),X: A,L: nat] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),L))
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qk(fun(nat,A),fun(nat,fun(nat,A)),X6),L),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_4559_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N8: nat] :
        ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R3),aa(nat,real,X6,N2))) )
     => filterlim(nat,real,aTP_Lamp_ql(fun(nat,real),fun(nat,real),X6),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_4560_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
     => filterlim(nat,real,aTP_Lamp_qm(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_4561_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K7: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K7))
              & ! [X2: A] : pp(aa(real,bool,aa(real,fun(real,bool),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)),K7))) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_4562_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,N2)),aa(nat,real,F2,aa(nat,nat,suc,N2))))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F2,N2)),L))
       => ( ! [E2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
             => ? [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,N5)),E2))) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_4563_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_qn(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_4564_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_4565_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_qo(fun(nat,A),fun(nat,A),F2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_4566_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_qp(fun(nat,A),fun(nat,A),F2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_4567_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_qq(real,fun(real,fun(nat,real)),X),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_4568_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real)))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_4569_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real)))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),C2)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_4570_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => filterlim(nat,real,aTP_Lamp_qr(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_4571_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,F2,S)
        <=> filterlim(nat,A,aTP_Lamp_qs(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def'
tff(fact_4572_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),X: real] :
          ( filterlim(nat,real,aTP_Lamp_qt(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,X),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_4573_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => ? [No: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N5))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N5)),L5))),R2)) ) ) ) ) ).

% LIMSEQ_D
tff(fact_4574_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No2),N2))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N2)),L5))),R3)) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_4575_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No3: nat] :
                ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N3)),L5))),R5)) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_4576_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_4577_tendsto__at__iff__sequentially,axiom,
    ! [C: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,C),A2: C,X: A,S: set(A)] :
          ( filterlim(A,C,F2,topolo7230453075368039082e_nhds(C,A2),topolo174197925503356063within(A,X,S))
        <=> ! [X8: fun(nat,A)] :
              ( ! [I4: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,X8,I4)),aa(set(A),set(A),aa(set(A),fun(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))))))
             => ( filterlim(nat,A,X8,topolo7230453075368039082e_nhds(A,X),at_top(nat))
               => filterlim(nat,C,aa(fun(nat,A),fun(nat,C),comp(A,C,nat,F2),X8),topolo7230453075368039082e_nhds(C,A2),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_4578_tendsto__power__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(B,nat),F4: filter(B),X: A] :
          ( filterlim(B,nat,F2,at_top(nat),F4)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
           => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_qu(fun(B,nat),fun(A,fun(B,A)),F2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_4579_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & ? [E4: fun(A,B)] :
                ( ! [H6: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H6)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F9,H6))),aa(A,B,E4,H6))
                & filterlim(A,real,aTP_Lamp_qv(fun(A,B),fun(A,real),E4),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% has_derivative_iff_Ex
tff(fact_4580_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),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_4581_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,S: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N2: nat] : aa(nat,A,S,N2) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_qw(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),S),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_4582_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,aTP_Lamp_qx(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_4583_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X)))
         => filterlim(nat,A,aTP_Lamp_qy(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_4584_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D6: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,D6,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D6)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_qz(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D6),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_4585_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ra(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F9),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_at_within
tff(fact_4586_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F9: fun(A,B),X: A,F2: fun(A,B),S: 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_rb(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F9),X),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S))
           => has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivativeI
tff(fact_4587_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F9,F4)
        <=> ( 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_rd(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F9),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_4588_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [X: A,S2: set(A),F2: fun(A,B),F9: fun(A,B)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
         => ( topolo1002775350975398744n_open(A,S2)
           => ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,S2))
            <=> ( real_V3181309239436604168linear(A,B,F9)
                & ? [E4: fun(A,B)] :
                    ( ! [H6: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H6)),S2))
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H6)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F9,H6))),aa(A,B,E4,H6)) ) )
                    & filterlim(A,real,aTP_Lamp_qv(fun(A,B),fun(A,real),E4),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ) ).

% has_derivative_at_within_iff_Ex
tff(fact_4589_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E3: real,F9: fun(A,B),S: set(A),X: A,F2: fun(A,B),H7: fun(A,real)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
         => ( real_V3181309239436604168linear(A,B,F9)
           => ( ! [Y2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),S))
                 => ( ( Y2 != X )
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y2,X)),E3))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),divide_divide(real,real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,Y2)),aa(A,B,F2,X))),aa(A,B,F9,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y2),X)))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y2),X)))),aa(A,real,H7,Y2))) ) ) )
             => ( filterlim(A,real,H7,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S))
               => has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_4590_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_re(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_4591_open__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1002775350975398744n_open(A,bot_bot(set(A))) ) ).

% open_empty
tff(fact_4592_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V557655796197034286t_dist(A,zero_zero(A),X) = real_V7770717601297561774m_norm(A,X) ) ).

% dist_0_norm
tff(fact_4593_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y)))
        <=> ( X != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_4594_open__Inter,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(set(A))] :
          ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),S2))
         => ( ! [X4: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),S2))
               => topolo1002775350975398744n_open(A,X4) )
           => topolo1002775350975398744n_open(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S2)) ) ) ) ).

% open_Inter
tff(fact_4595_open__INT,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: set(B),B3: fun(B,set(A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => topolo1002775350975398744n_open(A,aa(B,set(A),B3,X4)) )
           => topolo1002775350975398744n_open(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A3))) ) ) ) ).

% open_INT
tff(fact_4596_openI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
             => ? [T9: set(A)] :
                  ( topolo1002775350975398744n_open(A,T9)
                  & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),T9))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),S2)) ) )
         => topolo1002775350975398744n_open(A,S2) ) ) ).

% openI
tff(fact_4597_open__subopen,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
             => ? [T10: set(A)] :
                  ( topolo1002775350975398744n_open(A,T10)
                  & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),T10))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T10),S2)) ) ) ) ) ).

% open_subopen
tff(fact_4598_first__countable__basis,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
        ? [A5: fun(nat,set(A))] :
          ( ! [I3: nat] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(nat,set(A),A5,I3)))
              & topolo1002775350975398744n_open(A,aa(nat,set(A),A5,I3)) )
          & ! [S9: set(A)] :
              ( ( topolo1002775350975398744n_open(A,S9)
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S9)) )
             => ? [I2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),A5,I2)),S9)) ) ) ) ).

% first_countable_basis
tff(fact_4599_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X: A,E3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y,X)),E3))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),E3)) ) ) ).

% dist_commute_lessI
tff(fact_4600_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,D2: real] : topolo1002775350975398744n_open(A,aa(fun(A,bool),set(A),collect(A),aa(real,fun(A,bool),aTP_Lamp_rf(A,fun(real,fun(A,bool)),X),D2))) ) ).

% open_ball
tff(fact_4601_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
             => ? [E4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
                  & ! [Y4: A] :
                      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y4,X3)),E4))
                     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S2)) ) ) ) ) ) ).

% open_dist
tff(fact_4602_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V7770717601297561774m_norm(A,X) = real_V557655796197034286t_dist(A,X,zero_zero(A)) ) ).

% norm_conv_dist
tff(fact_4603_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),zero_zero(real))) ) ).

% dist_not_less_zero
tff(fact_4604_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y))) ) ) ).

% dist_pos_lt
tff(fact_4605_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) )
           => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Sup_Sup(A),A3)),A3)) ) ) ) ).

% Sup_notin_open
tff(fact_4606_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) )
           => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Inf_Inf(A),A3)),A3)) ) ) ) ).

% Inf_notin_open
tff(fact_4607_not__open__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [X: A] : ~ topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ).

% not_open_singleton
tff(fact_4608_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A,E3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z))),E3))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),E3)) ) ) ).

% dist_triangle_lt
tff(fact_4609_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Y: A,E1: real,X22: A,E22: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X1,Y)),E1))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),E22))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X1,X22)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% dist_triangle_less_add
tff(fact_4610_separation__t2,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ? [U4: set(A),V4: set(A)] :
              ( topolo1002775350975398744n_open(A,U4)
              & topolo1002775350975398744n_open(A,V4)
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),U4))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),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_4611_hausdorff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ? [U5: set(A),V5: set(A)] :
              ( topolo1002775350975398744n_open(A,U5)
              & topolo1002775350975398744n_open(A,V5)
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),U5))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),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_4612_at__within__open__subset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S2: set(A),T3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S2))
         => ( topolo1002775350975398744n_open(A,S2)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),T3))
             => ( topolo174197925503356063within(A,A2,T3) = topolo174197925503356063within(A,A2,top_top(set(A))) ) ) ) ) ) ).

% at_within_open_subset
tff(fact_4613_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
             => ? [B4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B4))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,X,B4)),S2)) ) ) ) ) ) ).

% open_right
tff(fact_4614_Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S: set(A)] :
          ( ( topolo174197925503356063within(A,X,S) != bot_bot(filter(A)) )
         => ( topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_rg(A,A)) = X ) ) ) ).

% Lim_ident_at
tff(fact_4615_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F9: A,A2: A,S2: set(A),D2: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F9,topolo174197925503356063within(A,A2,S2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S2))
             => ( ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D2))
                     => ( aa(A,A,F2,X4) = aa(A,A,G,X4) ) ) )
               => has_field_derivative(A,G,F9,topolo174197925503356063within(A,A2,S2)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_4616_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),X: A,S: set(A),D2: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,X,S))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
             => ( ! [X9: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X9),S))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X9,X)),D2))
                     => ( aa(A,B,F2,X9) = aa(A,B,G,X9) ) ) )
               => has_derivative(A,B,G,F9,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_4617_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M3))
                 => ! [N3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N3))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M3),aa(nat,A,X6,N3))),E4)) ) ) ) ) ) ).

% Cauchy_def
tff(fact_4618_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [N7: nat] :
                ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S,N3),aa(nat,A,S,N7))),E4)) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_4619_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
           => ? [M8: nat] :
              ! [M2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M2))
               => ! [N5: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N5))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M2),aa(nat,A,X6,N5))),E3)) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_4620_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
             => ? [M10: nat] :
                ! [M5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M5))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M5),aa(nat,A,X6,N2))),E2)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% metric_CauchyI
tff(fact_4621_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)
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),F0),S10))
               => ? [N7: nat] :
                  ! [N3: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N3))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,F2,N3)),S10)) ) ) ) ) ) ).

% lim_explicit
tff(fact_4622_not__tendsto__and__filterlim__at__infinity,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F4: filter(B),F2: fun(B,A),C2: A] :
          ( ( F4 != bot_bot(filter(B)) )
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
           => ~ filterlim(B,A,F2,at_infinity(A),F4) ) ) ) ).

% not_tendsto_and_filterlim_at_infinity
tff(fact_4623_tendsto__Lim,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Net: filter(A),F2: fun(A,B),L: B] :
          ( ( Net != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),Net)
           => ( topolo3827282254853284352ce_Lim(A,B,Net,F2) = L ) ) ) ) ).

% tendsto_Lim
tff(fact_4624_filterlim__pow__at__top,axiom,
    ! [A: $tType,N: nat,F2: fun(A,real),F4: filter(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_oa(nat,fun(fun(A,real),fun(A,real)),N),F2),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_4625_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Y: A,E3: real,X22: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X1,Y)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X1,X22)),E3)) ) ) ) ).

% dist_triangle_half_l
tff(fact_4626_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X1: A,E3: real,X22: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y,X1)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y,X22)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X1,X22)),E3)) ) ) ) ).

% dist_triangle_half_r
tff(fact_4627_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,X: A,S2: set(A),D2: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
           => ( ! [X9: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X9),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X9,X)))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X9,X)),D2))
                     => ( aa(A,B,F2,X9) = aa(A,B,G,X9) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% Lim_transform_within
tff(fact_4628_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,X22: A,E3: real,X32: A,X42: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X1,X22)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X22,X32)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X1,X42)),E3)) ) ) ) ) ).

% dist_triangle_third
tff(fact_4629_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A,S2: set(A),T3: set(A),U2: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
         => ( topolo1002775350975398744n_open(A,S2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T3),S2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),S2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,X,T3) = topolo174197925503356063within(A,X,U2) ) ) ) ) ) ).

% at_within_nhd
tff(fact_4630_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G5: filter(B),X: A,S2: set(A),F4: filter(B),D2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G5,topolo174197925503356063within(A,X,S2))
         => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),G5),F4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
             => ( ! [X9: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X9),S2))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X9,X)))
                     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X9,X)),D2))
                       => ( aa(A,B,F2,X9) = aa(A,B,G,X9) ) ) ) )
               => filterlim(A,B,F2,F4,topolo174197925503356063within(A,X,S2)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_4631_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M3))
                 => ! [N3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N3))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F2,M3),aa(nat,A,F2,N3))),E4)) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_4632_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
             => ? [M10: nat] :
                ! [M5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M5))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M5),aa(nat,A,X6,N2))),E2)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI'
tff(fact_4633_at__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] :
          ( ( topolo174197925503356063within(A,A2,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),A2),bot_bot(set(A)))) ) ) ).

% at_eq_bot_iff
tff(fact_4634_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_4635_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_4636_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ri(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_4637_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,F2: fun(A,real),F4: filter(A)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),S),zero_zero(real)))
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rj(real,fun(fun(A,real),fun(A,real)),S),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_neg_powr
tff(fact_4638_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [S7: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S7))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),S7)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X3),L5)),R5)) ) ) ) ) ) ).

% LIM_def
tff(fact_4639_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => ? [S4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S4))
                & ! [X2: A] :
                    ( ( ( X2 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),S4)) )
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X2),L5)),R2)) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_4640_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [S8: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S8))
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),S8)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L5)),R3)) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_4641_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A2: A,R: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
           => ( ! [X4: A] :
                  ( ( X4 != A2 )
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),R))
                   => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_4642_tendsto__mult__filterlim__at__infinity,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(B,A),C2: A,F4: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( ( C2 != zero_zero(A) )
           => ( filterlim(B,A,G,at_infinity(A),F4)
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rk(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),at_infinity(A),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_4643_tendsto__divide__0,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(C,A),C2: A,F4: filter(C),G: fun(C,A)] :
          ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( filterlim(C,A,G,at_infinity(A),F4)
           => filterlim(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_rl(fun(C,A),fun(fun(C,A),fun(C,A)),F2),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_4644_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),N: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_rm(fun(A,B),fun(nat,fun(A,B)),F2),N),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_4645_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No3: nat] :
                ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N3),L5)),R5)) ) ) ) ) ).

% lim_sequentially
tff(fact_4646_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No2),N2))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N2),L5)),R3)) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_4647_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
           => ? [No: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N5))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N5),L5)),R2)) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_4648_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [J3: nat] :
            ? [M9: nat] :
            ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M3))
             => ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M3),aa(nat,A,X6,N3))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_4649_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D4))
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D4)) )
                     => ( aa(A,B,F2,X4) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_rn(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_4650_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_4651_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_ro(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F4)
        <=> filterlim(A,B,G,at_infinity(B),F4) ) ) ).

% filterlim_inverse_at_iff
tff(fact_4652_tendsto__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A2: A,S2: set(A),F2: fun(A,D),L5: D] :
          ( nO_MATCH(C,A,zero_zero(C),A2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S2))
           => ( topolo1002775350975398744n_open(A,S2)
             => ( filterlim(A,D,F2,topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,A2,S2))
              <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_pd(A,fun(fun(A,D),fun(A,D)),A2),F2),topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_4653_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),No3))
                  & ! [N3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N3))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N3),L5)),R5)) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_4654_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F4: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_mr(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_4655_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),X4))
         => ? [Y3: real] :
              ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X4,top_top(set(real))))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),zero_zero(real))) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Flim),aa(real,real,F2,B2))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_4656_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X)))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),X),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_4657_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_4658_totally__bounded__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [K3: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite2(A),K3))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),aTP_Lamp_rq(real,fun(A,set(A)),E4)),K3)))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_4659_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,N: nat,B3: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
             => eventually(A,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_rr(fun(nat,A),fun(nat,fun(real,fun(A,bool))),C2),N),B3),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_4660_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F2: fun(real,real),F4: filter(real)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_rs(nat,fun(fun(real,real),fun(real,real)),N),F2),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_4661_totally__bounded__empty,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => topolo6688025880775521714ounded(A,bot_bot(set(A))) ) ).

% totally_bounded_empty
tff(fact_4662_eventually__const,axiom,
    ! [A: $tType,F4: filter(A),P: bool] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_md(bool,fun(A,bool),P),F4)
      <=> pp(P) ) ) ).

% eventually_const
tff(fact_4663_totally__bounded__Union,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [M4: set(set(A))] :
          ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),M4))
         => ( ! [S6: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),S6),M4))
               => topolo6688025880775521714ounded(A,S6) )
           => topolo6688025880775521714ounded(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),M4)) ) ) ) ).

% totally_bounded_Union
tff(fact_4664_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_rt(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F4) ) ) ).

% filterlim_at_bot
tff(fact_4665_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_rt(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_4666_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_ru(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_4667_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_rv(fun(A,real),fun(A,bool),F2),F4)
       => filterlim(A,real,F2,at_bot(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_4668_eventually__bot,axiom,
    ! [A: $tType,P: fun(A,bool)] : eventually(A,P,bot_bot(filter(A))) ).

% eventually_bot
tff(fact_4669_eventually__happens,axiom,
    ! [A: $tType,P: fun(A,bool),Net: filter(A)] :
      ( eventually(A,P,Net)
     => ( ( Net = bot_bot(filter(A)) )
        | ? [X_1: A] : pp(aa(A,bool,P,X_1)) ) ) ).

% eventually_happens
tff(fact_4670_eventually__happens_H,axiom,
    ! [A: $tType,F4: filter(A),P: fun(A,bool)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,P,F4)
       => ? [X_1: A] : pp(aa(A,bool,P,X_1)) ) ) ).

% eventually_happens'
tff(fact_4671_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,bool),Net: filter(A)] :
      ( ! [X4: A] : ~ pp(aa(A,bool,P,X4))
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_4672_eventually__const__iff,axiom,
    ! [A: $tType,P: bool,F4: filter(A)] :
      ( eventually(A,aTP_Lamp_md(bool,fun(A,bool),P),F4)
    <=> ( pp(P)
        | ( F4 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_4673_trivial__limit__def,axiom,
    ! [A: $tType,F4: filter(A)] :
      ( ( F4 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_ao(A,bool),F4) ) ).

% trivial_limit_def
tff(fact_4674_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N7: A] :
            ! [N3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N3),N7))
             => pp(aa(A,bool,P,N3)) ) ) ) ).

% eventually_at_bot_dense
tff(fact_4675_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_rw(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_4676_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),aTP_Lamp_rx(A,fun(A,bool)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_4677_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N7: A] :
            ! [N3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N3),N7))
             => pp(aa(A,bool,P,N3)) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_4678_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_ry(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_4679_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,bool)] :
          ( ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),X4))
             => pp(aa(A,bool,P,X4)) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_4680_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N7: A] :
            ! [N3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N7),N3))
             => pp(aa(A,bool,P,N3)) ) ) ) ).

% eventually_at_top_linorder
tff(fact_4681_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N7: A] :
            ! [N3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N7),N3))
             => pp(aa(A,bool,P,N3)) ) ) ) ).

% eventually_at_top_dense
tff(fact_4682_eventually__sequentially,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N7: nat] :
        ! [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N3))
         => pp(aa(nat,bool,P,N3)) ) ) ).

% eventually_sequentially
tff(fact_4683_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,bool)] :
      ( ! [X4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),X4))
         => pp(aa(nat,bool,P,X4)) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_4684_trivial__limit__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_bot(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_bot_linorder
tff(fact_4685_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),ord_less_eq(A),C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_4686_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),ord_less(A),C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_4687_le__sequentially,axiom,
    ! [F4: filter(nat)] :
      ( pp(aa(filter(nat),bool,aa(filter(nat),fun(filter(nat),bool),ord_less_eq(filter(nat)),F4),at_top(nat)))
    <=> ! [N7: nat] : eventually(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),F4) ) ).

% le_sequentially
tff(fact_4688_totally__bounded__subset,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A),T3: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),S2))
           => topolo6688025880775521714ounded(A,T3) ) ) ) ).

% totally_bounded_subset
tff(fact_4689_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_rv(fun(A,real),fun(A,bool),F2),F4)
       => filterlim(A,real,aTP_Lamp_rz(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ) ).

% filterlim_inverse_at_bot
tff(fact_4690_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),top_top(A)))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B7: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B7),top_top(A)))
                & ! [Z2: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B7),Z2))
                   => pp(aa(A,bool,P,Z2)) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_4691_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A)] :
          ( ! [X4: A,Y2: A] :
              ( pp(aa(A,bool,Q,X4))
             => ( pp(aa(A,bool,Q,Y2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y2))) ) ) )
         => ( ! [X4: B] :
                ( pp(aa(B,bool,P,X4))
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( pp(aa(B,bool,P,X4))
                 => pp(aa(A,bool,Q,aa(B,A,G,X4))) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_4692_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,X: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_lessThan(A,X)))
          <=> ? [B7: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B7),X))
                & ! [Y4: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B7),Y4))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
                     => pp(aa(A,bool,P,Y4)) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_4693_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,bool),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_lessThan(A,X)))
        <=> ? [B7: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B7),X))
              & ! [Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B7),Y4))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
                   => pp(aa(A,bool,P,Y4)) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_4694_tendsto__sandwich,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),G: fun(B,A),Net: filter(B),H: fun(B,A),C2: A] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_sa(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),Net)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_sa(fun(B,A),fun(fun(B,A),fun(B,bool)),G),H),Net)
           => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),Net)
             => ( filterlim(B,A,H,topolo7230453075368039082e_nhds(A,C2),Net)
               => filterlim(B,A,G,topolo7230453075368039082e_nhds(A,C2),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_4695_order__tendsto__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),X: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F4)
        <=> ( ! [L4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L4),X))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_sb(fun(B,A),fun(A,fun(B,bool)),F2),L4),F4) )
            & ! [U6: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),U6))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_sc(fun(B,A),fun(A,fun(B,bool)),F2),U6),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_4696_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F4: filter(B)] :
          ( ! [A4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A4),Y))
             => eventually(B,aa(A,fun(B,bool),aTP_Lamp_sb(fun(B,A),fun(A,fun(B,bool)),F2),A4),F4) )
         => ( ! [A4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A4))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_sc(fun(B,A),fun(A,fun(B,bool)),F2),A4),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F4) ) ) ) ).

% order_tendstoI
tff(fact_4697_order__tendstoD_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),Y: A,F4: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F4)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),Y))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_sb(fun(B,A),fun(A,fun(B,bool)),F2),A2),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_4698_order__tendstoD_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),Y: A,F4: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F4)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A2))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_sc(fun(B,A),fun(A,fun(B,bool)),F2),A2),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_4699_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_sd(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F4) ) ) ).

% filterlim_at_top
tff(fact_4700_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_sd(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_4701_filterlim__at__top__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),F4: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F2,at_top(A),F4)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_se(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),F4)
           => filterlim(B,A,G,at_top(A),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_4702_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_sf(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_4703_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_sg(fun(A,real),fun(A,bool),F2),F4)
       => filterlim(A,real,F2,at_top(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_4704_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
          ~ ! [A5: fun(nat,set(A))] :
              ( ! [I3: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A5,I3))
             => ( ! [I3: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(nat,set(A),A5,I3)))
               => ~ ! [S9: set(A)] :
                      ( topolo1002775350975398744n_open(A,S9)
                     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S9))
                       => eventually(nat,aa(set(A),fun(nat,bool),aTP_Lamp_sh(fun(nat,set(A)),fun(set(A),fun(nat,bool)),A5),S9),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_4705_eventually__Inf__base,axiom,
    ! [A: $tType,B3: set(filter(A)),P: fun(A,bool)] :
      ( ( B3 != bot_bot(set(filter(A))) )
     => ( ! [F5: filter(A)] :
            ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),F5),B3))
           => ! [G4: filter(A)] :
                ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),G4),B3))
               => ? [X2: filter(A)] :
                    ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),X2),B3))
                    & pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),X2),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),G4))) ) ) )
       => ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3))
        <=> ? [X3: filter(A)] :
              ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),X3),B3))
              & eventually(A,P,X3) ) ) ) ) ).

% eventually_Inf_base
tff(fact_4706_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,bool),F4: fun(A,filter(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),A3)))
      <=> ? [Q7: fun(A,fun(B,bool))] :
            ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => eventually(B,aa(A,fun(B,bool),Q7,X3),aa(A,filter(B),F4,X3)) )
            & ! [Y4: B] :
                ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q7,X3),Y4)) )
               => pp(aa(B,bool,P,Y4)) ) ) ) ) ).

% eventually_INF_finite
tff(fact_4707_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),B2),A2))
     => eventually(real,aa(real,fun(real,bool),aTP_Lamp_si(real,fun(real,fun(real,bool)),B2),A2),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ).

% eventually_at_left_real
tff(fact_4708_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A,S2: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S2))
        <=> ? [D5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
              & ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                 => ( ( ( X3 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D5)) )
                   => pp(aa(A,bool,P,X3)) ) ) ) ) ) ).

% eventually_at
tff(fact_4709_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
              & ! [X3: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D5))
                 => pp(aa(A,bool,P,X3)) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_4710_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),set_or5935395276787703475ssThan(A,A2,B2)))
             => pp(aa(A,bool,P,X4)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => eventually(A,P,topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_4711_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,bool),A2: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,bool),aTP_Lamp_sj(fun(A,bool),fun(A,fun(A,bool)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_4712_decreasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [L: A,F2: fun(B,A),F4: filter(B)] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_sk(A,fun(fun(B,A),fun(B,bool)),L),F2),F4)
         => ( ! [X4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),X4))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_sc(fun(B,A),fun(A,fun(B,bool)),F2),X4),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_4713_increasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F2: fun(B,A),L: A,F4: filter(B)] :
          ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_sl(fun(B,A),fun(A,fun(B,bool)),F2),L),F4)
         => ( ! [X4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),L))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_sb(fun(B,A),fun(A,fun(B,bool)),F2),X4),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_4714_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_sm(fun(A,B),fun(B,fun(A,bool)),F2),Z7),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_4715_tendsto__le,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F4: filter(B),F2: fun(B,A),X: A,G: fun(B,A),Y: A] :
          ( ( F4 != bot_bot(filter(B)) )
         => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F4)
           => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,Y),F4)
             => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_sn(fun(B,A),fun(fun(B,A),fun(B,bool)),F2),G),F4)
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ) ) ).

% tendsto_le
tff(fact_4716_tendsto__lowerbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(B,A),X: A,F4: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F4)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_so(fun(B,A),fun(A,fun(B,bool)),F2),A2),F4)
           => ( ( F4 != bot_bot(filter(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X)) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_4717_tendsto__upperbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(B,A),X: A,F4: filter(B),A2: A] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,X),F4)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_sp(fun(B,A),fun(A,fun(B,bool)),F2),A2),F4)
           => ( ( F4 != bot_bot(filter(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2)) ) ) ) ) ).

% tendsto_upperbound
tff(fact_4718_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F4: fun(B,filter(A)),B3: set(B)] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),B3)))
    <=> ? [X8: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X8),B3))
          & pp(aa(set(B),bool,finite_finite2(B),X8))
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),X8))) ) ) ).

% eventually_INF
tff(fact_4719_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A,S2: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S2))
        <=> ? [D5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
              & ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                 => ( ( ( X3 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X3,A2)),D5)) )
                   => pp(aa(A,bool,P,X3)) ) ) ) ) ) ).

% eventually_at_le
tff(fact_4720_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_infinity(A))
        <=> ? [B7: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B7))
              & ! [X3: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X3)))
                 => pp(aa(A,bool,P2,X3)) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_4721_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,bool),aTP_Lamp_sq(fun(A,B),fun(B,fun(A,bool)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_lessThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_4722_tendsto__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_sr(fun(B,A),fun(A,fun(real,fun(B,bool))),F2),L),E4),F4) ) ) ) ).

% tendsto_iff
tff(fact_4723_tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F4: filter(B)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
             => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_sr(fun(B,A),fun(A,fun(real,fun(B,bool))),F2),L),E2),F4) )
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ).

% tendstoI
tff(fact_4724_tendstoD,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(B,A),L: A,F4: filter(B),E3: real] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
           => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_sr(fun(B,A),fun(A,fun(real,fun(B,bool))),F2),L),E3),F4) ) ) ) ).

% tendstoD
tff(fact_4725_eventually__Inf,axiom,
    ! [A: $tType,P: fun(A,bool),B3: set(filter(A))] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3))
    <=> ? [X8: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X8),B3))
          & pp(aa(set(filter(A)),bool,finite_finite2(filter(A)),X8))
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X8)) ) ) ).

% eventually_Inf
tff(fact_4726_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => ( eventually(A,aTP_Lamp_sg(fun(A,real),fun(A,bool),G),F4)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ss(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_4727_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_sg(fun(A,real),fun(A,bool),F2),F4)
     => ( filterlim(A,real,F2,at_top(real),F4)
      <=> filterlim(A,real,aa(fun(A,real),fun(A,real),comp(real,real,A,inverse_inverse(real)),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_4728_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_sg(fun(A,real),fun(A,bool),F2),F4)
       => filterlim(A,real,aTP_Lamp_rz(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ) ).

% filterlim_inverse_at_top
tff(fact_4729_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_sg(fun(A,real),fun(A,bool),F2),F4)
     => ( filterlim(A,real,aTP_Lamp_rz(fun(A,real),fun(A,real),F2),at_top(real),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_4730_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A),A2: A] :
          ( ! [X4: A,Y2: A] :
              ( pp(aa(A,bool,Q,X4))
             => ( pp(aa(A,bool,Q,Y2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y2))) ) ) )
         => ( ! [X4: B] :
                ( pp(aa(B,bool,P,X4))
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( pp(aa(B,bool,P,X4))
                 => pp(aa(A,bool,Q,aa(B,A,G,X4))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)))
               => ( ! [B4: A] :
                      ( pp(aa(A,bool,Q,B4))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),A2)) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_4731_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B3: set(A),F4: fun(A,filter(B)),P: fun(B,bool)] :
      ( ( B3 != bot_bot(set(A)) )
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),B3))
           => ! [B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B3))
               => ? [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B3))
                    & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F4,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,A4)),aa(A,filter(B),F4,B4)))) ) ) )
       => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B3)))
        <=> ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3))
              & eventually(B,P,aa(A,filter(B),F4,X3)) ) ) ) ) ).

% eventually_INF_base
tff(fact_4732_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,C),K4: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_st(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),F2),G),K4),F4)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_4733_filterlim__at__withinI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),C2: A,F4: filter(B),A3: set(A)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( eventually(B,aa(set(A),fun(B,bool),aa(A,fun(set(A),fun(B,bool)),aTP_Lamp_su(fun(B,A),fun(A,fun(set(A),fun(B,bool))),F2),C2),A3),F4)
           => filterlim(B,A,F2,topolo174197925503356063within(A,C2,A3),F4) ) ) ) ).

% filterlim_at_withinI
tff(fact_4734_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_bot(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ri(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_4735_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ri(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_4736_filterlim__at__infinity,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: real,F2: fun(C,A),F4: filter(C)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C2))
         => ( filterlim(C,A,F2,at_infinity(A),F4)
          <=> ! [R5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),R5))
               => eventually(C,aa(real,fun(C,bool),aTP_Lamp_sv(fun(C,A),fun(real,fun(C,bool)),F2),R5),F4) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_4737_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_sw(fun(A,real),fun(A,bool),F2),F4)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sx(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_4738_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_sw(fun(A,real),fun(A,bool),F2),F4)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sx(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ) ).

% tendsto_powr2
tff(fact_4739_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( ( ( A2 != zero_zero(real) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
              & eventually(A,aTP_Lamp_sw(fun(A,real),fun(A,bool),F2),F4) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sx(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ).

% tendsto_powr'
tff(fact_4740_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2))
         => ? [Y3: real] :
              ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X4,top_top(set(real))))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Flim),aa(real,real,F2,B2))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_4741_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F2: fun(real,real),F4: filter(real)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_rs(nat,fun(fun(real,real),fun(real,real)),N),F2),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_4742_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_sy(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_Cauchy'
tff(fact_4743_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [Y4: B,K5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K5))
              & eventually(A,aa(real,fun(A,bool),aa(B,fun(real,fun(A,bool)),aTP_Lamp_sz(fun(A,B),fun(B,fun(real,fun(A,bool))),F2),Y4),K5),F4) ) ) ) ).

% Bfun_metric_def
tff(fact_4744_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
         => ~ ! [B8: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B8))
               => ~ eventually(A,aa(real,fun(A,bool),aTP_Lamp_ta(fun(A,B),fun(real,fun(A,bool)),F2),B8),F4) ) ) ) ).

% BfunE
tff(fact_4745_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_tb(fun(A,bool),fun(A,bool),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_4746_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),D2: B] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B3))
       => pp(aa(set(fun(A,B)),bool,finite_finite2(fun(A,B)),aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aa(B,fun(fun(A,B),bool),aa(set(B),fun(B,fun(fun(A,B),bool)),aTP_Lamp_tc(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),A3),B3),D2)))) ) ) ).

% finite_set_of_finite_funs
tff(fact_4747_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eg(A,fun(fun(nat,A),fun(nat,A)),C2),F2),at_top(nat))
          <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_4748_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
         => ? [K7: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K7))
              & ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),K7)) ) ) ) ).

% BseqD
tff(fact_4749_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
         => ~ ! [K7: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K7))
               => ~ ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),K7)) ) ) ) ).

% BseqE
tff(fact_4750_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K4: real,X6: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K4))
         => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N2))),K4))
           => bfun(nat,A,X6,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_4751_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K5))
              & ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N3))),K5)) ) ) ) ).

% Bseq_def
tff(fact_4752_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N7: nat] :
            ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N3))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% Bseq_iff1a
tff(fact_4753_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K3))
              & ? [N7: nat] :
                ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N3)),aa(A,A,uminus_uminus(A),aa(nat,A,X6,N7))))),K3)) ) ) ) ).

% Bseq_iff3
tff(fact_4754_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K3))
              & ? [X3: A] :
                ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N3)),aa(A,A,uminus_uminus(A),X3)))),K3)) ) ) ) ).

% Bseq_iff2
tff(fact_4755_Bfun__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A2: A,F4: filter(B)] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( A2 != zero_zero(A) )
           => bfun(B,A,aTP_Lamp_oo(fun(B,A),fun(B,A),F2),F4) ) ) ) ).

% Bfun_inverse
tff(fact_4756_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [K5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K5))
              & eventually(A,aa(real,fun(A,bool),aTP_Lamp_ta(fun(A,B),fun(real,fun(A,bool)),F2),K5),F4) ) ) ) ).

% Bfun_def
tff(fact_4757_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_td(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_bounded_partials
tff(fact_4758_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] : order_Greatest(A,P) = the(A,aTP_Lamp_te(fun(A,bool),fun(A,bool),P)) ) ).

% Greatest_def
tff(fact_4759_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y5: set(B),X6: set(A),F4: filter(B),F2: fun(A,C)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),Y5)),X6))
     => ( eventually(B,aTP_Lamp_tf(set(B),fun(B,bool),Y5),F4)
       => ( map_filter_on(A,C,X6,F2,map_filter_on(B,A,Y5,G,F4)) = map_filter_on(B,C,Y5,aa(fun(B,A),fun(B,C),comp(A,C,B,F2),G),F4) ) ) ) ).

% map_filter_on_comp
tff(fact_4760_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_or3652927894154168847AtMost(nat,L,U))) ).

% finite_greaterThanAtMost
tff(fact_4761_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_or3652927894154168847AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),U)) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_4762_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_4763_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_4764_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_4765_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or3652927894154168847AtMost(A,A2,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% infinite_Ioc_iff
tff(fact_4766_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_4767_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_4768_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_4769_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_4770_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C2,D2) )
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),C2)) )
            | ( ( A2 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_4771_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_4772_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or3652927894154168847AtMost(A,A2,B2))) ) ) ).

% infinite_Ioc
tff(fact_4773_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_4774_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,bool),B2: nat] :
      ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
     => ( ! [Y2: nat] :
            ( pp(aa(nat,bool,P,Y2))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y2),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_ex_nat
tff(fact_4775_Greatest__le__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y2: nat] :
            ( pp(aa(nat,bool,P,Y2))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y2),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),order_Greatest(nat,P))) ) ) ).

% Greatest_le_nat
tff(fact_4776_GreatestI__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y2: nat] :
            ( pp(aa(nat,bool,P,Y2))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y2),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_nat
tff(fact_4777_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)) = bot_bot(set(A)) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),C2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),A2)) ) ) ) ).

% Ioc_disjoint
tff(fact_4778_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
             => ? [B4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),X))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B4,X)),S2)) ) ) ) ) ) ).

% open_left
tff(fact_4779_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_4780_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)),set_ord_atMost(A,L)),set_or3652927894154168847AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(3)
tff(fact_4781_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_4782_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y2: A] :
                ( pp(aa(A,bool,P,Y2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) )
           => ( order_Greatest(A,P) = X ) ) ) ) ).

% Greatest_equality
tff(fact_4783_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y2: A] :
                ( pp(aa(A,bool,P,Y2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) )
           => ( ! [X4: A] :
                  ( pp(aa(A,bool,P,X4))
                 => ( ! [Y3: A] :
                        ( pp(aa(A,bool,P,Y3))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X4)) )
                   => pp(aa(A,bool,Q,X4)) ) )
             => pp(aa(A,bool,Q,order_Greatest(A,P))) ) ) ) ) ).

% GreatestI2_order
tff(fact_4784_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% sum.head
tff(fact_4785_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% prod.head
tff(fact_4786_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_4787_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_4788_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_4789_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or3652927894154168847AtMost(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_4790_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_4791_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F4: filter(A)] :
          ( topolo6773858410816713723filter(A,F4)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [P5: fun(A,bool)] :
                  ( eventually(A,P5,F4)
                  & ! [X3: A,Y4: A] :
                      ( ( pp(aa(A,bool,P5,X3))
                        & pp(aa(A,bool,P5,Y4)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,Y4)),E4)) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_4792_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),P: fun(A,bool)] : aa(fun(A,bool),A,least(A,Less_eq),P) = the(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_tg(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Less_eq),P)) ).

% ord.Least_def
tff(fact_4793_eventually__filtercomap__at__topological,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,bool),F2: fun(A,B),A3: B,B3: set(B)] :
          ( eventually(A,P,filtercomap(A,B,F2,topolo174197925503356063within(B,A3,B3)))
        <=> ? [S10: set(B)] :
              ( topolo1002775350975398744n_open(B,S10)
              & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S10))
              & ! [X3: A] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X3)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S10),B3)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))))
                 => pp(aa(A,bool,P,X3)) ) ) ) ) ).

% eventually_filtercomap_at_topological
tff(fact_4794_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or3652927894154168847AtMost(int,L,U))) ).

% finite_greaterThanAtMost_int
tff(fact_4795_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_4796_ord_OLeast_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : least(A,Less_eq) = least(A,Less_eq) ).

% ord.Least.cong
tff(fact_4797_filtercomap__neq__bot,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F2: fun(B,A)] :
      ( ! [P6: fun(A,bool)] :
          ( eventually(A,P6,F4)
         => ? [X2: B] : pp(aa(A,bool,P6,aa(B,A,F2,X2))) )
     => ( filtercomap(B,A,F2,F4) != bot_bot(filter(B)) ) ) ).

% filtercomap_neq_bot
tff(fact_4798_eventually__filtercomap__at__top__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),F2: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F2,at_top(A)))
        <=> ? [N7: A] :
            ! [X3: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N7),aa(B,A,F2,X3)))
             => pp(aa(B,bool,P,X3)) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_4799_eventually__filtercomap__at__top__dense,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(B,bool),F2: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F2,at_top(A)))
        <=> ? [N7: A] :
            ! [X3: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N7),aa(B,A,F2,X3)))
             => pp(aa(B,bool,P,X3)) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_4800_filtercomap__neq__bot__surj,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F2: fun(B,A)] :
      ( ( F4 != 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,F4) != bot_bot(filter(B)) ) ) ) ).

% filtercomap_neq_bot_surj
tff(fact_4801_eventually__filtercomap__at__bot__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),F2: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F2,at_bot(A)))
        <=> ? [N7: A] :
            ! [X3: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),N7))
             => pp(aa(B,bool,P,X3)) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_4802_eventually__filtercomap__at__bot__dense,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(B,bool),F2: fun(B,A)] :
          ( eventually(B,P,filtercomap(B,A,F2,at_bot(A)))
        <=> ? [N7: A] :
            ! [X3: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X3)),N7))
             => pp(aa(B,bool,P,X3)) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_4803_lex__prod__def,axiom,
    ! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_ti(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_4804_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A,S: set(A)] : topolo174197925503356063within(A,X,S) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(set(A)),set(filter(A)),image2(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_tj(A,fun(set(A),fun(set(A),filter(A))),X),S)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_tk(A,fun(set(A),bool),X)))) ) ).

% at_within_eq
tff(fact_4805_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K4: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gi(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K4)))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_my(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_4806_principal__le__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),principal(A,A3)),principal(A,B3)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% principal_le_iff
tff(fact_4807_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B,R2: set(product_prod(A,A)),S: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6))),lex_prod(A,B,R2,S)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A6)),R2))
        | ( ( A2 = A6 )
          & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B6)),S)) ) ) ) ).

% in_lex_prod
tff(fact_4808_continuous__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B)] : topolo3448309680560233919inuous(A,B,bot_bot(filter(A)),F2) ) ).

% continuous_bot
tff(fact_4809_continuous__trivial__limit,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Net: filter(A),F2: fun(A,B)] :
          ( ( Net = bot_bot(filter(A)) )
         => topolo3448309680560233919inuous(A,B,Net,F2) ) ) ).

% continuous_trivial_limit
tff(fact_4810_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_tl(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% isCont_Pair
tff(fact_4811_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,C,F4,G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F4,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_tl(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_Pair
tff(fact_4812_bot__eq__principal__empty,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = principal(A,bot_bot(set(A))) ).

% bot_eq_principal_empty
tff(fact_4813_principal__eq__bot__iff,axiom,
    ! [A: $tType,X6: set(A)] :
      ( ( principal(A,X6) = bot_bot(filter(A)) )
    <=> ( X6 = bot_bot(set(A)) ) ) ).

% principal_eq_bot_iff
tff(fact_4814_IVT,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),A2: A,Y: B,B2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,A2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F2,B2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( ! [X4: A] :
                    ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X4,top_top(set(A))),F2) )
               => ? [X4: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2))
                    & ( aa(A,B,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_4815_IVT2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),B2: A,Y: B,A2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,B2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F2,A2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( ! [X4: A] :
                    ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X4,top_top(set(A))),F2) )
               => ? [X4: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2))
                    & ( aa(A,B,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_4816_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aa(fun(A,B),fun(A,B),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_4817_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_tn(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_4818_nhds__discrete,axiom,
    ! [A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [X: A] : topolo7230453075368039082e_nhds(A,X) = principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ).

% nhds_discrete
tff(fact_4819_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_to(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_4820_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_rg(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_4821_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_rg(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_tn(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_4822_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_rg(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_to(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_4823_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X4: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M8: A] :
              ! [X2: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X2))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F2,X2)),M8)) ) ) ) ) ).

% isCont_bounded
tff(fact_4824_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X4: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X2: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X2))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F2,X2)),M8)) )
                & ? [X4: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2))
                    & ( aa(real,A,F2,X4) = M8 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_4825_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X4: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X2: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X2))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M8),aa(real,A,F2,X2))) )
                & ? [X4: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2))
                    & ( aa(real,A,F2,X4) = M8 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_4826_isCont__inverse__function2,axiom,
    ! [A2: real,X: real,B2: real,G: fun(real,real),F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ( ! [Z3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z3),B2))
               => ( aa(real,real,G,aa(real,real,F2,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z3),B2))
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_4827_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_4828_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_to(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_4829_tendsto__principal__singleton,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(B,A),X: B] : filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,aa(B,A,F2,X)),principal(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B))))) ) ).

% tendsto_principal_singleton
tff(fact_4830_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),F2)
         => ( ( cos(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_ou(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_4831_nhds__discrete__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A] :
          ( topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
         => ( topolo7230453075368039082e_nhds(A,X) = principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ) ) ).

% nhds_discrete_open
tff(fact_4832_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),F2)
         => ( ( sin(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_ov(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_4833_continuous__at__within__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: C,A3: set(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A3),F2)
         => ( ( cosh(A,aa(C,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A3),aTP_Lamp_tp(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_4834_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( cos(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tq(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_ou(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_4835_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tq(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_ov(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_4836_continuous__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(C),F2: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F4,F2)
         => ( ( cosh(A,aa(C,A,F2,topolo3827282254853284352ce_Lim(C,C,F4,aTP_Lamp_tr(C,C)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,F4,aTP_Lamp_tp(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_tanh
tff(fact_4837_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X4: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X2: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X2))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F2,X2)),M8)) )
                & ! [N8: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N8),M8))
                   => ? [X4: real] :
                        ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N8),aa(real,A,F2,X4))) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_4838_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_rg(A,A)))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_ts(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_4839_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_4840_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_4841_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_4842_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( cos(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_ou(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_4843_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_4844_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),G)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,A2)))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,A2)))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),aa(fun(A,real),fun(A,real),aTP_Lamp_tt(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_4845_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( sin(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_ov(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_4846_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,real,F4,G)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_rg(A,A)))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_rg(A,A))) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_rg(A,A)))))
                 => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_tt(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_4847_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [X: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tu(A,fun(fun(A,B),fun(A,B)),X),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,X)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_4848_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I6: set(A),F4: fun(A,set(B)),F2: fun(B,C),G5: fun(D,set(C)),J4: set(D)] :
      ( ( I6 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
           => ! [J2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I6))
               => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F4,I2)),aa(A,set(B),F4,J2)))
                  | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F4,J2)),aa(A,set(B),F4,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_tv(fun(D,set(C)),fun(D,filter(C)),G5)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_tw(fun(A,set(B)),fun(A,filter(B)),F4)),I6)))
        <=> ! [X3: D] :
              ( pp(aa(set(D),bool,aa(D,fun(set(D),bool),member(D),X3),J4))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),I6))
                  & ! [Xb4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xb4),aa(A,set(B),F4,Xa3)))
                     => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,F2,Xb4)),aa(D,set(C),G5,X3))) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_4849_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D6: real,G: fun(real,real),X: real,A2: real,B2: real] :
      ( has_field_derivative(real,F2,D6,topolo174197925503356063within(real,aa(real,real,G,X),top_top(set(real))))
     => ( ( D6 != zero_zero(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
           => ( ! [Y2: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Y2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y2),B2))
                   => ( aa(real,real,F2,aa(real,real,G,Y2)) = Y2 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D6),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_4850_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_4851_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_4852_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F2: fun(A,set(B))] :
      ( pp(aa(set(A),bool,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),aTP_Lamp_tw(fun(A,set(B)),fun(A,filter(B)),F2)),X6)) = principal(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),X6))) ) ) ).

% INF_principal_finite
tff(fact_4853_LIM__less__bound,axiom,
    ! [B2: real,X: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),B2),X))
     => ( ! [X4: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),set_or5935395276787703475ssThan(real,B2,X)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X4))) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X))) ) ) ) ).

% LIM_less_bound
tff(fact_4854_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,A2)))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,A2)))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_tt(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_4855_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_4856_at__within__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S: set(A)] : topolo174197925503356063within(A,A2,S) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A2)),principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ).

% at_within_def
tff(fact_4857_isCont__inverse__function,axiom,
    ! [D2: real,X: real,G: fun(real,real),F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
     => ( ! [Z3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z3),X))),D2))
           => ( aa(real,real,G,aa(real,real,F2,Z3)) = Z3 ) )
       => ( ! [Z3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z3),X))),D2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_4858_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real),G6: fun(real,real),F9: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [Z3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z3),B2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F2) ) )
       => ( ! [Z3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z3),B2))
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),G) ) )
         => ( ! [Z3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),B2))
                 => has_field_derivative(real,G,aa(real,real,G6,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
           => ( ! [Z3: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z3))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),B2))
                   => has_field_derivative(real,F2,aa(real,real,F9,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
             => ? [C4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),C4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C4),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,G6,C4)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A2))),aa(real,real,F9,C4)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_4859_metric__isCont__LIM__compose2,axiom,
    ! [D: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(D) )
     => ! [A2: A,F2: fun(A,C),G: fun(C,D),L: D] :
          ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(C,D,G,topolo7230453075368039082e_nhds(D,L),topolo174197925503356063within(C,aa(A,C,F2,A2),top_top(set(C))))
           => ( ? [D4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D4))
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D4)) )
                     => ( aa(A,C,F2,X4) != aa(A,C,F2,A2) ) ) )
             => filterlim(A,D,aa(fun(C,D),fun(A,D),aTP_Lamp_tx(fun(A,C),fun(fun(C,D),fun(A,D)),F2),G),topolo7230453075368039082e_nhds(D,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_4860_at__left__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( topolo174197925503356063within(A,X,set_ord_lessThan(A,X)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_ty(A,fun(A,filter(A)),X)),set_ord_lessThan(A,X))) ) ) ) ).

% at_left_eq
tff(fact_4861_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D4))
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A2))),D4)) )
                     => ( aa(A,B,F2,X4) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_pg(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_4862_isCont__powser_H,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,Aa),C2: fun(nat,Aa),K4: Aa] :
          ( topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( summable(Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_tz(fun(nat,Aa),fun(Aa,fun(nat,Aa)),C2),K4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(Aa,aa(A,Aa,F2,A2))),real_V7770717601297561774m_norm(Aa,K4)))
             => topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_ub(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_4863_complete__uniform,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A)] :
          ( topolo2479028161051973599mplete(A,S2)
        <=> ! [F10: filter(A)] :
              ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F10),principal(A,S2)))
             => ( ( F10 != bot_bot(filter(A)) )
               => ( topolo6773858410816713723filter(A,F10)
                 => ? [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                      & pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F10),topolo7230453075368039082e_nhds(A,X3))) ) ) ) ) ) ) ).

% complete_uniform
tff(fact_4864_at__within__order,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,S: set(A)] :
          ( ( top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
         => ( topolo174197925503356063within(A,X,S) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_uc(A,fun(set(A),fun(A,filter(A))),X),S)),aa(A,set(A),set_ord_greaterThan(A),X)))),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_ud(A,fun(set(A),fun(A,filter(A))),X),S)),set_ord_lessThan(A,X)))) ) ) ) ).

% at_within_order
tff(fact_4865_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),R: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_uf(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),P),R)))) ).

% same_fst_def
tff(fact_4866_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),aa(A,set(A),set_ord_greaterThan(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),I)) ) ) ).

% greaterThan_iff
tff(fact_4867_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),X)),aa(A,set(A),set_ord_greaterThan(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% greaterThan_subset_iff
tff(fact_4868_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),top_top(A)))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),X)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_4869_same__fstI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),X: A,Y6: B,Y: B,R: fun(A,set(product_prod(B,B)))] :
      ( pp(aa(A,bool,P,X))
     => ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y6),Y)),aa(A,set(product_prod(B,B)),R,X)))
       => pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y6)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y))),same_fst(A,B,P,R))) ) ) ).

% same_fstI
tff(fact_4870_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_4871_infinite__Ioi,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A2: A] : ~ pp(aa(set(A),bool,finite_finite2(A),aa(A,set(A),set_ord_greaterThan(A),A2))) ) ).

% infinite_Ioi
tff(fact_4872_greaterThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] : aa(A,set(A),set_ord_greaterThan(A),X) != bot_bot(set(A)) ) ).

% greaterThan_non_empty
tff(fact_4873_trivial__limit__at__right__real,axiom,
    ! [A: $tType] :
      ( ( dense_order(A)
        & no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)) != bot_bot(filter(A)) ) ).

% trivial_limit_at_right_real
tff(fact_4874_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_4875_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,Y: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)))
          <=> ? [B7: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B7))
                & ! [Y4: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),B7))
                     => pp(aa(A,bool,P,Y4)) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_4876_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,bool),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)))
        <=> ? [B7: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B7))
              & ! [Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),B7))
                   => pp(aa(A,bool,P,Y4)) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_4877_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( topolo174197925503356063within(A,A2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)) ) ) ) ).

% at_within_Icc_at_right
tff(fact_4878_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_4879_trivial__limit__at__right__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ( topolo174197925503356063within(A,top_top(A),aa(A,set(A),set_ord_greaterThan(A),top_top(A))) = bot_bot(filter(A)) ) ) ).

% trivial_limit_at_right_top
tff(fact_4880_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_4881_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : eventually(A,aa(A,fun(A,bool),ord_less(A),X),topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X))) ) ).

% eventually_at_right_less
tff(fact_4882_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ? [A4: A,B4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),set_ord_lessThan(A,A4)))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(A,set(A),set_ord_greaterThan(A),B4)))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,A4)),aa(A,set(A),set_ord_greaterThan(A),B4)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_4883_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),set_or5935395276787703475ssThan(A,A2,B2)))
             => pp(aa(A,bool,P,X4)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% eventually_at_rightI
tff(fact_4884_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_4885_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => eventually(real,aa(real,fun(real,bool),aTP_Lamp_si(real,fun(real,fun(real,bool)),A2),B2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ).

% eventually_at_right_real
tff(fact_4886_greaterThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),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_4887_filterlim__times__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(B,A),P2: A,F12: filter(B),C2: A,L: A] :
          ( filterlim(B,A,F2,topolo174197925503356063within(A,P2,aa(A,set(A),set_ord_greaterThan(A),P2)),F12)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( ( L = aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2) )
             => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_ug(fun(B,A),fun(A,fun(B,A)),F2),C2),topolo174197925503356063within(A,L,aa(A,set(A),set_ord_greaterThan(A),L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_4888_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,bool),aTP_Lamp_uh(fun(A,B),fun(B,fun(A,bool)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_greaterThan(B),L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_4889_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F2: fun(A,B),P: fun(B,bool),G: fun(B,A),A2: A] :
          ( ! [X4: A,Y2: A] :
              ( pp(aa(A,bool,Q,X4))
             => ( pp(aa(A,bool,Q,Y2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y2))) ) ) )
         => ( ! [X4: B] :
                ( pp(aa(B,bool,P,X4))
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( pp(aa(B,bool,P,X4))
                 => pp(aa(A,bool,Q,aa(B,A,G,X4))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
               => ( ! [B4: A] :
                      ( pp(aa(A,bool,Q,B4))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B4)) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_4890_nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A] : topolo7230453075368039082e_nhds(A,X) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(real),set(filter(A)),image2(real,filter(A),aTP_Lamp_uj(A,fun(real,filter(A)),X)),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% nhds_metric
tff(fact_4891_at__right__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_uk(A,fun(A,filter(A)),X)),aa(A,set(A),set_ord_greaterThan(A),X))) ) ) ) ).

% at_right_eq
tff(fact_4892_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,G: fun(A,B),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ul(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_4893_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A)] :
          ( ! [A4: A,B4: A,X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),S2))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),S2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X4))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B4))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2)) ) ) ) )
         => ? [A4: A,B4: A] :
              ( ( S2 = bot_bot(set(A)) )
              | ( S2 = top_top(set(A)) )
              | ( S2 = set_ord_lessThan(A,B4) )
              | ( S2 = set_ord_atMost(A,B4) )
              | ( S2 = aa(A,set(A),set_ord_greaterThan(A),A4) )
              | ( S2 = set_ord_atLeast(A,A4) )
              | ( S2 = set_or5935395276787703475ssThan(A,A4,B4) )
              | ( S2 = set_or3652927894154168847AtMost(A,A4,B4) )
              | ( S2 = set_or7035219750837199246ssThan(A,A4,B4) )
              | ( S2 = set_or1337092689740270186AtMost(A,A4,B4) ) ) ) ) ).

% interval_cases
tff(fact_4894_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( ! [F3: fun(nat,A)] :
                ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(nat,A,F3,N5)))
               => ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N5)),B2))
                 => ( order_antimono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_um(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_4895_GMVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X4: real] :
            ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
       => ( ! [X4: real] :
              ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2)) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X4,top_top(set(real)))) )
         => ( ! [X4: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X4),B2)) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X4,top_top(set(real))),G) )
           => ( ! [X4: real] :
                  ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2)) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X4,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C4: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),C4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C4),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_4896_atLeast__0,axiom,
    set_ord_atLeast(nat,zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_4897_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),set_ord_atLeast(A,K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),I)) ) ) ).

% atLeast_iff
tff(fact_4898_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_4899_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,X)),set_ord_atLeast(A,Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% atLeast_subset_iff
tff(fact_4900_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atLeast(A,L2)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L2),L)) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_4901_differentiable__cmult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [C2: A,Q4: fun(B,A),T2: B] :
          ( differentiable(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_un(A,fun(fun(B,A),fun(B,A)),C2),Q4),topolo174197925503356063within(B,T2,top_top(set(B))))
        <=> ( ( C2 = zero_zero(A) )
            | differentiable(B,A,Q4,topolo174197925503356063within(B,T2,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_4902_differentiable__cmult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [Q4: fun(B,A),C2: A,T2: B] :
          ( differentiable(B,A,aa(A,fun(B,A),aTP_Lamp_uo(fun(B,A),fun(A,fun(B,A)),Q4),C2),topolo174197925503356063within(B,T2,top_top(set(B))))
        <=> ( ( C2 = zero_zero(A) )
            | differentiable(B,A,Q4,topolo174197925503356063within(B,T2,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_4903_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_4904_infinite__Ici,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A2: A] : ~ pp(aa(set(A),bool,finite_finite2(A),set_ord_atLeast(A,A2))) ) ).

% infinite_Ici
tff(fact_4905_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_atLeast(A,L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_4906_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,Y4)),aa(A,B,F2,X3))) ) ) ) ).

% antimono_def
tff(fact_4907_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,Y2)),aa(A,B,F2,X4))) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_4908_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,X))) ) ) ) ).

% antimonoE
tff(fact_4909_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,X))) ) ) ) ).

% antimonoD
tff(fact_4910_atLeast__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] :
          ( ( set_ord_atLeast(A,X) = top_top(set(A)) )
        <=> ( X = bot_bot(A) ) ) ) ).

% atLeast_eq_UNIV_iff
tff(fact_4911_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_ord_atLeast(A,L))) ) ).

% not_UNIV_le_Ici
tff(fact_4912_differentiable__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X: A,S: set(A),T2: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => differentiable(A,B,F2,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% differentiable_within_subset
tff(fact_4913_not__Ici__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,L2: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,L)),set_or1337092689740270186AtMost(A,L2,H2))) ) ).

% not_Ici_le_Icc
tff(fact_4914_not__Iic__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,H)),set_ord_atLeast(A,L2))) ) ).

% not_Iic_le_Ici
tff(fact_4915_not__Ici__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,L)),set_ord_atMost(A,H2))) ) ).

% not_Ici_le_Iic
tff(fact_4916_Ioi__le__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),A2)),set_ord_atLeast(A,A2))) ) ).

% Ioi_le_Ico
tff(fact_4917_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A3)
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,aa(nat,nat,suc,I))),aa(nat,A,A3,I))) ) ) ).

% decseq_SucD
tff(fact_4918_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N2))),aa(nat,A,X6,N2)))
         => order_antimono(nat,A,X6) ) ) ).

% decseq_SucI
tff(fact_4919_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N3))),aa(nat,A,F2,N3))) ) ) ).

% decseq_Suc_iff
tff(fact_4920_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( order_antimono(nat,A,F2)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,J)),aa(nat,A,F2,I))) ) ) ) ).

% decseqD
tff(fact_4921_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( order_antimono(nat,A,X6)
        <=> ! [M3: nat,N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N3)),aa(nat,A,X6,M3))) ) ) ) ).

% decseq_def
tff(fact_4922_differentiable__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [S: set(A),F2: fun(A,fun(B,C)),Net: filter(B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
               => differentiable(B,C,aa(A,fun(B,C),F2,X4),Net) )
           => differentiable(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_uq(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2),Net) ) ) ) ).

% differentiable_sum
tff(fact_4923_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,A2)),aa(A,set(A),set_ord_greaterThan(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_4924_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_4925_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_4926_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ur(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% differentiable_divide
tff(fact_4927_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,B,F2,X) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_us(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% differentiable_inverse
tff(fact_4928_atMost__Int__atLeast,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [N: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atMost(A,N)),set_ord_atLeast(A,N)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),N),bot_bot(set(A))) ) ).

% atMost_Int_atLeast
tff(fact_4929_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L5: A,N: nat] :
          ( order_antimono(nat,A,X6)
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L5),aa(nat,A,X6,N))) ) ) ) ).

% decseq_ge
tff(fact_4930_atLeast__Suc,axiom,
    ! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atLeast(nat,K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_4931_tendsto__at__right__sequentially,axiom,
    ! [C: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: B,B2: B,X6: fun(B,C),L5: C] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A2),B2))
         => ( ! [S6: fun(nat,B)] :
                ( ! [N5: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A2),aa(nat,B,S6,N5)))
               => ( ! [N5: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(nat,B,S6,N5)),B2))
                 => ( order_antimono(nat,B,S6)
                   => ( filterlim(nat,B,S6,topolo7230453075368039082e_nhds(B,A2),at_top(nat))
                     => filterlim(nat,C,aa(fun(nat,B),fun(nat,C),aTP_Lamp_ut(fun(B,C),fun(fun(nat,B),fun(nat,C)),X6),S6),topolo7230453075368039082e_nhds(C,L5),at_top(nat)) ) ) ) )
           => filterlim(B,C,X6,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,aa(B,set(B),set_ord_greaterThan(B),A2))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_4932_Gcd__eq__Max,axiom,
    ! [M4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M4))
     => ( ( M4 != bot_bot(set(nat)) )
       => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M4))
         => ( gcd_Gcd(nat,M4) = 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_uu(nat,set(nat))),M4))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_4933_continuous__at__Sup__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( order_antimono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Sup_Sup(A),S2),set_ord_lessThan(A,aa(set(A),A,complete_Sup_Sup(A),S2))),F2)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S2)
               => ( aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),S2)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),S2)) ) ) ) ) ) ) ).

% continuous_at_Sup_antimono
tff(fact_4934_continuous__at__Inf__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( order_antimono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Inf_Inf(A),S2),aa(A,set(A),set_ord_greaterThan(A),aa(set(A),A,complete_Inf_Inf(A),S2))),F2)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S2)
               => ( aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),S2)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),S2)) ) ) ) ) ) ) ).

% continuous_at_Inf_antimono
tff(fact_4935_bdd__belowI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A),M: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X4)) )
         => condit1013018076250108175_below(A,A3) ) ) ).

% bdd_belowI
tff(fact_4936_bdd__below_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A),M4: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M4),X4)) )
         => condit1013018076250108175_below(A,A3) ) ) ).

% bdd_below.I
tff(fact_4937_bdd__above_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A),M4: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M4)) )
         => condit941137186595557371_above(A,A3) ) ) ).

% bdd_above.I
tff(fact_4938_bdd__below__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit1013018076250108175_below(A,bot_bot(set(A))) ) ).

% bdd_below_empty
tff(fact_4939_bdd__above__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit941137186595557371_above(A,bot_bot(set(A))) ) ).

% bdd_above_empty
tff(fact_4940_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% Max_singleton
tff(fact_4941_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_bd(nat,fun(nat,bool),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_4942_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),X)) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_4943_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),X)) ) ) ) ) ) ).

% Max_less_iff
tff(fact_4944_Max__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [A3: set(B),C2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( A3 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_uv(A,fun(B,A),C2)),A3)) = C2 ) ) ) ) ).

% Max_const
tff(fact_4945_bdd__below__UN,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [I6: set(B),A3: fun(B,set(A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( condit1013018076250108175_below(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),I6)))
          <=> ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I6))
               => condit1013018076250108175_below(A,aa(B,set(A),A3,X3)) ) ) ) ) ).

% bdd_below_UN
tff(fact_4946_bdd__above__UN,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [I6: set(B),A3: fun(B,set(A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),I6))
         => ( condit941137186595557371_above(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),I6)))
          <=> ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I6))
               => condit941137186595557371_above(A,aa(B,set(A),A3,X3)) ) ) ) ) ).

% bdd_above_UN
tff(fact_4947_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ).

% Max_insert
tff(fact_4948_bdd__above__nat,axiom,
    ! [X6: set(nat)] :
      ( condit941137186595557371_above(nat,X6)
    <=> pp(aa(set(nat),bool,finite_finite2(nat),X6)) ) ).

% bdd_above_nat
tff(fact_4949_bdd__above__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => condit941137186595557371_above(A,A3) ) ) ).

% bdd_above_finite
tff(fact_4950_bdd__below__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => condit1013018076250108175_below(A,A3) ) ) ).

% bdd_below_finite
tff(fact_4951_bdd__below__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: set(A),A3: set(A)] :
          ( condit1013018076250108175_below(A,B3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
           => condit1013018076250108175_below(A,A3) ) ) ) ).

% bdd_below_mono
tff(fact_4952_bdd__below_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit1013018076250108175_below(A,A3)
        <=> ? [M9: A] :
            ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M9),X3)) ) ) ) ).

% bdd_below.unfold
tff(fact_4953_bdd__below_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit1013018076250108175_below(A,A3)
         => ~ ! [M8: A] :
                ~ ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M8),X2)) ) ) ) ).

% bdd_below.E
tff(fact_4954_bdd__above_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit941137186595557371_above(A,A3)
        <=> ? [M9: A] :
            ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),M9)) ) ) ) ).

% bdd_above.unfold
tff(fact_4955_bdd__above_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit941137186595557371_above(A,A3)
         => ~ ! [M8: A] :
                ~ ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),M8)) ) ) ) ).

% bdd_above.E
tff(fact_4956_bdd__above__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: set(A),A3: set(A)] :
          ( condit941137186595557371_above(A,B3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
           => condit941137186595557371_above(A,A3) ) ) ) ).

% bdd_above_mono
tff(fact_4957_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A3)
           => ( condit1013018076250108175_below(A,A3)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ) ).

% cInf_le_cSup
tff(fact_4958_bdd__belowI2,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [A3: set(B),M: A,F2: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F2,X4))) )
         => condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3)) ) ) ).

% bdd_belowI2
tff(fact_4959_bdd__below_OI2,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [A3: set(B),M4: A,F2: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M4),aa(B,A,F2,X4))) )
         => condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3)) ) ) ).

% bdd_below.I2
tff(fact_4960_bdd__above_OI2,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [A3: set(B),F2: fun(B,A),M4: A] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),M4)) )
         => condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3)) ) ) ).

% bdd_above.I2
tff(fact_4961_cInf__lower,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( condit1013018076250108175_below(A,X6)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).

% cInf_lower
tff(fact_4962_cInf__lower2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A),Y: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( condit1013018076250108175_below(A,X6)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y)) ) ) ) ) ).

% cInf_lower2
tff(fact_4963_cSup__upper,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( condit941137186595557371_above(A,X6)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).

% cSup_upper
tff(fact_4964_cSup__upper2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A),Y: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
           => ( condit941137186595557371_above(A,X6)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ) ).

% cSup_upper2
tff(fact_4965_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A3))) ) ) ) ).

% Max.coboundedI
tff(fact_4966_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,finite_finite2(A),B3))
           => ( ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => ? [Xa: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),B3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa)) ) )
             => ( ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3))
                   => ? [Xa: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A3))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa)) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,lattic643756798349783984er_Max(A),B3) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_4967_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ! [Y2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = X ) ) ) ) ) ).

% Max_eqI
tff(fact_4968_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3))) ) ) ) ).

% Max_ge
tff(fact_4969_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),A3)) ) ) ) ).

% Max_in
tff(fact_4970_Max_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(A),A,lattic643756798349783984er_Max(A),A3) ) ) ) ) ).

% Max.in_idem
tff(fact_4971_cINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),X: B,U: A] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X)),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),U)) ) ) ) ) ).

% cINF_lower2
tff(fact_4972_cINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),X: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(B,A,F2,X))) ) ) ) ).

% cINF_lower
tff(fact_4973_le__cInf__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S2)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(set(A),A,complete_Inf_Inf(A),S2)))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3)) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_4974_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( ( B3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A3)
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B3))
                 => ? [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),B4)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ) ) ).

% cInf_mono
tff(fact_4975_cSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),X: B,U: A] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,X)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ) ) ).

% cSUP_upper2
tff(fact_4976_cSUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: B,A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ) ).

% cSUP_upper
tff(fact_4977_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X6)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y))
            <=> ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y)) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_4978_cSup__le__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S2)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S2)),A2))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A2)) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_4979_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( ( B3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A3)
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B3))
                 => ? [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),X2)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B3)),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ) ).

% cSup_mono
tff(fact_4980_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X6)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6)))
            <=> ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X3)) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_4981_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X)) ) ) ) ) ).

% Max.boundedI
tff(fact_4982_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X))
             => ! [A10: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A10),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A10),X)) ) ) ) ) ) ).

% Max.boundedE
tff(fact_4983_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A3) )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),M)) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4984_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)))
            <=> ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X3)) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_4985_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = M )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),M)) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4986_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)))
            <=> ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X3)) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_4987_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),A2)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_4988_cSup__eq__Max,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = aa(set(A),A,lattic643756798349783984er_Max(A),X6) ) ) ) ) ).

% cSup_eq_Max
tff(fact_4989_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ) ) ) ).

% Max_Sup
tff(fact_4990_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_4991_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(B,A,F2,I))) ) ) ) ) ).

% less_cINF_D
tff(fact_4992_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I: B] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),Y))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,I)),Y)) ) ) ) ) ).

% cSUP_lessD
tff(fact_4993_cINF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B3: set(B),F2: fun(C,A),A3: set(C),G: fun(B,A)] :
          ( ( B3 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,F2),A3))
           => ( ! [M5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),M5),B3))
                 => ? [X2: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),A3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F2,X2)),aa(B,A,G,M5))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),B3)))) ) ) ) ) ).

% cINF_mono
tff(fact_4994_le__cINF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),U: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))))
            <=> ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,X3))) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_4995_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,B3)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B3)),aa(set(A),A,complete_Inf_Inf(A),A3))) ) ) ) ) ).

% cInf_superset_mono
tff(fact_4996_cSUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),G: fun(C,A),B3: set(C),F2: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,G),B3))
           => ( ! [N2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N2),A3))
                 => ? [X2: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),B3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,N2)),aa(C,A,G,X2))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G),B3)))) ) ) ) ) ).

% cSUP_mono
tff(fact_4997_cSUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),U: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),U))
            <=> ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),U)) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_4998_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,B3)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ) ).

% cSup_subset_mono
tff(fact_4999_cInf__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( condit1013018076250108175_below(A,X6)
         => ( ( ( X6 = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X6)) = A2 ) )
            & ( ( X6 != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ) ) ).

% cInf_insert_If
tff(fact_5000_cInf__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X6)
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ) ).

% cInf_insert
tff(fact_5001_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B3))) ) ) ) ) ).

% Max.subset_imp
tff(fact_5002_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M4: set(A),N4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M4),N4))
         => ( ( M4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),N4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M4)),aa(set(A),A,lattic643756798349783984er_Max(A),N4))) ) ) ) ) ).

% Max_mono
tff(fact_5003_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N4: set(A)] :
          ( ! [X4: A,Y2: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X4),Y2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X4)),aa(A,A,H,Y2))
         => ( pp(aa(set(A),bool,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_5004_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( B3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B3)),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(A),A,lattic643756798349783984er_Max(A),A3) ) ) ) ) ) ).

% Max.subset
tff(fact_5005_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( A3 != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_5006_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X4: A,Y2: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X4),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),A3)) ) ) ) ) ).

% Max.closed
tff(fact_5007_cINF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: set(B),F2: fun(B,A),A2: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),A2))
            <=> ? [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X3)),A2)) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_5008_less__cSUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: set(B),F2: fun(B,A),A2: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))))
            <=> ? [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,X3))) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_5009_cINF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,G),A3))
             => ( 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,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),A3))) = 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_uw(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_5010_card__le__Suc__Max,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S2)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S2)))) ) ).

% card_le_Suc_Max
tff(fact_5011_Sup__nat__def,axiom,
    ! [X6: set(nat)] :
      ( ( ( X6 = bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = zero_zero(nat) ) )
      & ( ( X6 != bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),X6) ) ) ) ).

% Sup_nat_def
tff(fact_5012_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( divide_divide(nat,M,N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( divide_divide(nat,M,N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ux(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ) ).

% divide_nat_def
tff(fact_5013_Max__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S2: set(B),F2: fun(B,A),K: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( S2 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_uy(fun(B,A),fun(A,fun(B,A)),F2),K)),S2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image2(B,A,F2),S2))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_5014_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_uz(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_5015_cINF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),G: fun(B,A),B3: set(B),F2: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,G),B3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),B3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,G,X4)),aa(B,A,F2,X4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_5016_cSUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),G: fun(B,A),B3: set(B),F2: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G),B3))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,G,X4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),B3)))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_5017_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( condit1013018076250108175_below(A,A3)
         => ( condit1013018076250108175_below(A,B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) != bot_bot(set(A)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_5018_cINF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),A2: B] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( 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),A2),A3))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,A2)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))) ) ) ) ) ).

% cINF_insert
tff(fact_5019_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = 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),X),A3)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Max.insert_remove
tff(fact_5020_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Max.remove
tff(fact_5021_cINF__UNION,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(C),B3: fun(C,set(D)),F2: fun(D,B)] :
          ( ( A3 != bot_bot(set(C)) )
         => ( ! [X4: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),A3))
               => ( aa(C,set(D),B3,X4) != bot_bot(set(D)) ) )
           => ( condit1013018076250108175_below(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_va(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B3),F2)),A3)))
             => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,F2),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B3),A3)))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_vb(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B3),F2)),A3)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_5022_cSUP__UNION,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(C),B3: fun(C,set(D)),F2: fun(D,B)] :
          ( ( A3 != bot_bot(set(C)) )
         => ( ! [X4: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),A3))
               => ( aa(C,set(D),B3,X4) != bot_bot(set(D)) ) )
           => ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_va(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B3),F2)),A3)))
             => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,F2),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B3),A3)))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_vc(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B3),F2)),A3)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_5023_sum__le__card__Max,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image2(A,nat,F2),A3))))) ) ).

% sum_le_card_Max
tff(fact_5024_MVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2))
               => differentiable(real,real,F2,topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ? [L3: real,Z3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z3))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),B2))
              & has_field_derivative(real,F2,L3,topolo174197925503356063within(real,Z3,top_top(set(real))))
              & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),L3) ) ) ) ) ) ).

% MVT
tff(fact_5025_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_rx(A,fun(A,bool))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_5026_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_ve(real,filter(product_prod(A,A)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ).

% uniformity_dist
tff(fact_5027_continuous__on__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,C,S,G)
           => topolo81223032696312382ous_on(A,product_prod(B,C),S,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_vf(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_on_Pair
tff(fact_5028_uniformity__transE,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ~ ! [D7: fun(product_prod(A,A),bool)] :
                ( eventually(product_prod(A,A),D7,topolo7806501430040627800ormity(A))
               => ~ ! [X2: A,Y3: A] :
                      ( pp(aa(product_prod(A,A),bool,D7,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3)))
                     => ! [Z4: A] :
                          ( pp(aa(product_prod(A,A),bool,D7,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)))
                         => pp(aa(product_prod(A,A),bool,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z4))) ) ) ) ) ) ).

% uniformity_transE
tff(fact_5029_uniformity__trans,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ? [D7: fun(product_prod(A,A),bool)] :
              ( eventually(product_prod(A,A),D7,topolo7806501430040627800ormity(A))
              & ! [X2: A,Y3: A,Z4: A] :
                  ( pp(aa(product_prod(A,A),bool,D7,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3)))
                 => ( pp(aa(product_prod(A,A),bool,D7,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)))
                   => pp(aa(product_prod(A,A),bool,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z4))) ) ) ) ) ) ).

% uniformity_trans
tff(fact_5030_uniformity__refl,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool),X: A] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => pp(aa(product_prod(A,A),bool,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X))) ) ) ).

% uniformity_refl
tff(fact_5031_continuous__on__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(A),F2: fun(A,B),T2: set(A)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => topolo81223032696312382ous_on(A,B,T2,F2) ) ) ) ).

% continuous_on_subset
tff(fact_5032_uniformity__bot,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ( topolo7806501430040627800ormity(A) != bot_bot(filter(product_prod(A,A))) ) ) ).

% uniformity_bot
tff(fact_5033_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_5034_linorder_OMin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Min(A,Less_eq) = lattices_Min(A,Less_eq) ).

% linorder.Min.cong
tff(fact_5035_IVT2_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),B2: A,Y: B,A2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,B2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F2,A2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
               => ? [X4: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2))
                    & ( aa(A,B,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_5036_IVT_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F2: fun(A,B),A2: A,Y: B,B2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,A2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F2,B2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
               => ? [X4: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2))
                    & ( aa(A,B,F2,X4) = Y ) ) ) ) ) ) ) ).

% IVT'
tff(fact_5037_continuous__on__sing,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [X: A,F2: fun(A,B)] : topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),F2) ) ).

% continuous_on_sing
tff(fact_5038_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => ( ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
                 => ( aa(A,B,G,X4) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_vg(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_5039_continuous__on__compose2,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [T2: set(A),G: fun(A,B),S: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(A,B,T2,G)
         => ( topolo81223032696312382ous_on(C,A,S,F2)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F2),S)),T2))
             => topolo81223032696312382ous_on(C,B,S,aa(fun(C,A),fun(C,B),aTP_Lamp_vh(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2)) ) ) ) ) ).

% continuous_on_compose2
tff(fact_5040_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
               => ( aa(A,B,F2,X4) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_vi(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_5041_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
               => ( aa(A,B,F2,X4) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_vj(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_5042_continuous__onI__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & dense_order(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( topolo1002775350975398744n_open(B,aa(set(A),set(B),image2(A,B,F2),A3))
         => ( ! [X4: A,Y2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y2))) ) ) )
           => topolo81223032696312382ous_on(A,B,A3,F2) ) ) ) ).

% continuous_onI_mono
tff(fact_5043_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,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_vk(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_5044_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
               => ( cos(A,aa(A,A,F2,X4)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_ou(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_5045_open__Collect__less__Int,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ? [A5: set(A)] :
                ( topolo1002775350975398744n_open(A,A5)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),S) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,real),fun(A,bool),aa(fun(A,real),fun(fun(A,real),fun(A,bool)),aTP_Lamp_vl(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,bool))),S),F2),G)) ) ) ) ) ) ).

% open_Collect_less_Int
tff(fact_5046_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
               => ( sin(A,aa(A,A,F2,X4)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_ov(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_5047_continuous__on__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,A3,F2)
         => ( ! [X4: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),A3))
               => ( cosh(A,aa(C,A,F2,X4)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(C,A,A3,aTP_Lamp_vm(fun(C,A),fun(C,A),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_5048_uniformity__sym,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(A,A),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_vn(fun(product_prod(A,A),bool),fun(A,fun(A,bool)),E5)),topolo7806501430040627800ormity(A)) ) ) ).

% uniformity_sym
tff(fact_5049_open__Collect__positive,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ? [A5: set(A)] :
              ( topolo1002775350975398744n_open(A,A5)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),S) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,real),fun(A,bool),aTP_Lamp_vo(set(A),fun(fun(A,real),fun(A,bool)),S),F2)) ) ) ) ) ).

% open_Collect_positive
tff(fact_5050_continuous__on__powr_H,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [S: set(C),F2: fun(C,real),G: fun(C,real)] :
          ( topolo81223032696312382ous_on(C,real,S,F2)
         => ( topolo81223032696312382ous_on(C,real,S,G)
           => ( ! [X4: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),S))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(C,real,F2,X4)))
                    & ( ( aa(C,real,F2,X4) = zero_zero(real) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(C,real,G,X4))) ) ) )
             => topolo81223032696312382ous_on(C,real,S,aa(fun(C,real),fun(C,real),aTP_Lamp_vp(fun(C,real),fun(fun(C,real),fun(C,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_5051_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F2,X4))) )
             => ( ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
                   => ( aa(A,real,F2,X4) != one_one(real) ) )
               => ( ! [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X4))) )
                 => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_vq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_5052_Cauchy__uniform__iff,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [P5: fun(product_prod(A,A),bool)] :
              ( eventually(product_prod(A,A),P5,topolo7806501430040627800ormity(A))
             => ? [N7: nat] :
                ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N3))
                 => ! [M3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),M3))
                     => pp(aa(product_prod(A,A),bool,P5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,X6,N3)),aa(nat,A,X6,M3)))) ) ) ) ) ) ).

% Cauchy_uniform_iff
tff(fact_5053_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,B2: A,F2: fun(A,A)] :
          ( ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2))
               => ? [Y3: A] : has_field_derivative(A,F2,Y3,topolo174197925503356063within(A,X4,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_5054_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_vs(real,filter(product_prod(complex,complex)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_complex_def
tff(fact_5055_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_vu(real,filter(product_prod(real,real)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_real_def
tff(fact_5056_totally__bounded__def,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
        <=> ! [E6: fun(product_prod(A,A),bool)] :
              ( eventually(product_prod(A,A),E6,topolo7806501430040627800ormity(A))
             => ? [X8: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite2(A),X8))
                  & ! [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                     => ? [Xa3: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),X8))
                          & pp(aa(product_prod(A,A),bool,E6,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3))) ) ) ) ) ) ) ).

% totally_bounded_def
tff(fact_5057_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F9: fun(real,fun(real,real))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ( aa(real,real,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [X4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2))
                 => has_derivative(real,real,F2,aa(real,fun(real,real),F9,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
           => ? [Z3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z3))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),B2))
                & ! [X2: real] : aa(real,real,aa(real,fun(real,real),F9,Z3),X2) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_5058_mvt,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F9: fun(real,fun(real,real))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2))
               => has_derivative(real,real,F2,aa(real,fun(real,real),F9,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ~ ! [Xi: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Xi))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Xi),B2))
                 => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) != aa(real,real,aa(real,fun(real,real),F9,Xi),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)) ) ) ) ) ) ) ).

% mvt
tff(fact_5059_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_5060_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_5061_eventually__uniformity__metric,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [P: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),P,topolo7806501430040627800ormity(A))
        <=> ? [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
              & ! [X3: A,Y4: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,Y4)),E4))
                 => pp(aa(product_prod(A,A),bool,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4))) ) ) ) ) ).

% eventually_uniformity_metric
tff(fact_5062_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,A2)),aa(real,real,F2,B2))) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_5063_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),zero_zero(real))) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_5064_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ( aa(real,real,F2,B2) = aa(real,real,F2,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_5065_tendsto__iff__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E6: fun(product_prod(B,B),bool)] :
              ( eventually(product_prod(B,B),E6,topolo7806501430040627800ormity(B))
             => eventually(A,aa(fun(product_prod(B,B),bool),fun(A,bool),aa(B,fun(fun(product_prod(B,B),bool),fun(A,bool)),aTP_Lamp_vv(fun(A,B),fun(B,fun(fun(product_prod(B,B),bool),fun(A,bool))),F2),L),E6),F4) ) ) ) ).

% tendsto_iff_uniformity
tff(fact_5066_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),B2))
             => ( aa(real,real,F2,X) = aa(real,real,F2,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_5067_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,B2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)))
           => ( ! [X4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X4))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),B2))
                   => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X4)),topolo174197925503356063within(A,X4,top_top(set(A)))) ) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_5068_Rolle,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ( aa(real,real,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [X4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),B2))
                 => differentiable(real,real,F2,topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
           => ? [Z3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z3))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z3),B2))
                & has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_5069_compactE__image,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),C3: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ! [T6: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),T6),C3))
               => topolo1002775350975398744n_open(A,aa(B,set(A),F2,T6)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),C3))))
             => ~ ! [C8: set(B)] :
                    ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C8),C3))
                   => ( pp(aa(set(B),bool,finite_finite2(B),C8))
                     => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),C8)))) ) ) ) ) ) ) ).

% compactE_image
tff(fact_5070_compactE,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),T11: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T11)))
           => ( ! [B8: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B8),T11))
                 => topolo1002775350975398744n_open(A,B8) )
             => ~ ! [T12: set(set(A))] :
                    ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),T12),T11))
                   => ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),T12))
                     => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T12))) ) ) ) ) ) ) ).

% compactE
tff(fact_5071_compactI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( ! [C7: set(set(A))] :
              ( ! [X2: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X2),C7))
                 => topolo1002775350975398744n_open(A,X2) )
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C7)))
               => ? [C9: set(set(A))] :
                    ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C9),C7))
                    & pp(aa(set(set(A)),bool,finite_finite2(set(A)),C9))
                    & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C9))) ) ) )
         => topolo2193935891317330818ompact(A,S) ) ) ).

% compactI
tff(fact_5072_compact__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo2193935891317330818ompact(A,bot_bot(set(A))) ) ).

% compact_empty
tff(fact_5073_compact__attains__sup,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X4)) ) ) ) ) ) ).

% compact_attains_sup
tff(fact_5074_compact__attains__inf,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa)) ) ) ) ) ) ).

% compact_attains_inf
tff(fact_5075_continuous__attains__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F2)
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
                  & ! [Xa: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),S))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Xa))) ) ) ) ) ) ) ).

% continuous_attains_inf
tff(fact_5076_continuous__attains__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F2)
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
                  & ! [Xa: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),S))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,X4))) ) ) ) ) ) ) ).

% continuous_attains_sup
tff(fact_5077_compact__eq__Heine__Borel,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
        <=> ! [C6: set(set(A))] :
              ( ( ! [X3: set(A)] :
                    ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),C6))
                   => topolo1002775350975398744n_open(A,X3) )
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6))) )
             => ? [D8: set(set(A))] :
                  ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),D8),C6))
                  & pp(aa(set(set(A)),bool,finite_finite2(set(A)),D8))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),D8))) ) ) ) ) ).

% compact_eq_Heine_Borel
tff(fact_5078_uniformity__trans_H,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(product_prod(A,A),product_prod(A,A)),aa(fun(product_prod(A,A),fun(product_prod(A,A),bool)),fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),product_case_prod(product_prod(A,A),product_prod(A,A),bool),aa(fun(A,fun(A,fun(product_prod(A,A),bool))),fun(product_prod(A,A),fun(product_prod(A,A),bool)),product_case_prod(A,A,fun(product_prod(A,A),bool)),aTP_Lamp_vx(fun(product_prod(A,A),bool),fun(A,fun(A,fun(product_prod(A,A),bool))),E5))),prod_filter(product_prod(A,A),product_prod(A,A),topolo7806501430040627800ormity(A),topolo7806501430040627800ormity(A))) ) ) ).

% uniformity_trans'
tff(fact_5079_prod__filter__INF,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,I6: set(A),J4: set(B),A3: fun(A,filter(C)),B3: fun(B,filter(D))] :
      ( ( I6 != 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),A3),I6)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B3),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_vz(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J4),A3),B3)),I6)) ) ) ) ).

% prod_filter_INF
tff(fact_5080_prod__filter__INF1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I6: set(A),A3: fun(A,filter(B)),B3: filter(C)] :
      ( ( I6 != 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),A3),I6)),B3) = 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_wa(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A3),B3)),I6)) ) ) ).

% prod_filter_INF1
tff(fact_5081_prod__filter__mono__iff,axiom,
    ! [A: $tType,B: $tType,A3: filter(A),B3: filter(B),C3: filter(A),D6: filter(B)] :
      ( ( A3 != bot_bot(filter(A)) )
     => ( ( B3 != bot_bot(filter(B)) )
       => ( pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),prod_filter(A,B,A3,B3)),prod_filter(A,B,C3,D6)))
        <=> ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),A3),C3))
            & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),B3),D6)) ) ) ) ) ).

% prod_filter_mono_iff
tff(fact_5082_eventually__prod__sequentially,axiom,
    ! [P: fun(product_prod(nat,nat),bool)] :
      ( eventually(product_prod(nat,nat),P,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N7: nat] :
        ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),M3))
         => ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N3))
             => pp(aa(product_prod(nat,nat),bool,P,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N3),M3))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_5083_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,P: fun(product_prod(A,B),bool),F4: filter(A),G5: filter(B)] :
      ( eventually(product_prod(A,B),P,prod_filter(A,B,F4,G5))
    <=> ? [Pf: fun(A,bool),Pg: fun(B,bool)] :
          ( eventually(A,Pf,F4)
          & eventually(B,Pg,G5)
          & ! [X3: A,Y4: B] :
              ( pp(aa(A,bool,Pf,X3))
             => ( pp(aa(B,bool,Pg,Y4))
               => pp(aa(product_prod(A,B),bool,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4))) ) ) ) ) ).

% eventually_prod_filter
tff(fact_5084_eventually__prod__same,axiom,
    ! [A: $tType,P: fun(product_prod(A,A),bool),F4: filter(A)] :
      ( eventually(product_prod(A,A),P,prod_filter(A,A,F4,F4))
    <=> ? [Q7: fun(A,bool)] :
          ( eventually(A,Q7,F4)
          & ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,Q7,X3))
             => ( pp(aa(A,bool,Q7,Y4))
               => pp(aa(product_prod(A,A),bool,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4))) ) ) ) ) ).

% eventually_prod_same
tff(fact_5085_prod__filter__eq__bot,axiom,
    ! [A: $tType,B: $tType,A3: filter(A),B3: filter(B)] :
      ( ( prod_filter(A,B,A3,B3) = bot_bot(filter(product_prod(A,B))) )
    <=> ( ( A3 = bot_bot(filter(A)) )
        | ( B3 = bot_bot(filter(B)) ) ) ) ).

% prod_filter_eq_bot
tff(fact_5086_nhds__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [A2: A,B2: B] : topolo7230453075368039082e_nhds(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) = prod_filter(A,B,topolo7230453075368039082e_nhds(A,A2),topolo7230453075368039082e_nhds(B,B2)) ) ).

% nhds_prod
tff(fact_5087_filterlim__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G5: filter(B),F4: filter(A),G: fun(A,C),H7: filter(C)] :
      ( filterlim(A,B,F2,G5,F4)
     => ( filterlim(A,C,G,H7,F4)
       => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_wb(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),prod_filter(B,C,G5,H7),F4) ) ) ).

% filterlim_Pair
tff(fact_5088_eventually__prod1,axiom,
    ! [A: $tType,B: $tType,B3: filter(A),P: fun(B,bool),A3: filter(B)] :
      ( ( B3 != bot_bot(filter(A)) )
     => ( eventually(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),aTP_Lamp_wc(fun(B,bool),fun(B,fun(A,bool)),P)),prod_filter(B,A,A3,B3))
      <=> eventually(B,P,A3) ) ) ).

% eventually_prod1
tff(fact_5089_eventually__prod2,axiom,
    ! [A: $tType,B: $tType,A3: filter(A),P: fun(B,bool),B3: filter(B)] :
      ( ( A3 != bot_bot(filter(A)) )
     => ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_wd(fun(B,bool),fun(A,fun(B,bool)),P)),prod_filter(A,B,A3,B3))
      <=> eventually(B,P,B3) ) ) ).

% eventually_prod2
tff(fact_5090_prod__filter__INF2,axiom,
    ! [C: $tType,B: $tType,A: $tType,J4: set(A),A3: filter(B),B3: fun(A,filter(C))] :
      ( ( J4 != bot_bot(set(A)) )
     => ( prod_filter(B,C,A3,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),B3),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_we(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A3),B3)),J4)) ) ) ).

% prod_filter_INF2
tff(fact_5091_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: filter(A),G5: filter(B),H7: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F4,G5),H7) = 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_wg(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)))),prod_filter(A,product_prod(B,C),F4,prod_filter(B,C,G5,H7))) ).

% prod_filter_assoc
tff(fact_5092_sequentially__imp__eventually__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,A2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( ! [F3: fun(nat,A)] :
                ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(nat,A,F3,N5)))
               => ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N5)),A2))
                 => ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F3))
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_um(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_5093_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_wh(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_wi(set(product_prod(A,A)),fun(nat,bool),R)))))) ) ) ).

% relpow_finite_bounded1
tff(fact_5094_filtermap__bot,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : filtermap(B,A,F2,bot_bot(filter(B))) = bot_bot(filter(A)) ).

% filtermap_bot
tff(fact_5095_finite__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(set(product_prod(A,A)),bool,finite_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))),N),R))) ) ) ).

% finite_relpow
tff(fact_5096_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),F4: filter(C)] : pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),filtermap(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_wj(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G),F4)),prod_filter(A,B,filtermap(C,A,F2,F4),filtermap(C,B,G,F4)))) ).

% filtermap_Pair
tff(fact_5097_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mono_invE
tff(fact_5098_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => pp(aa(fun(A,A),bool,order_mono(A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2))) ) ) ).

% mono_pow
tff(fact_5099_mono__Suc,axiom,
    pp(aa(fun(nat,nat),bool,order_mono(nat,nat),suc)) ).

% mono_Suc
tff(fact_5100_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A] : pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),plus_plus(A),A2))) ) ).

% mono_add
tff(fact_5101_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mono_strict_invE
tff(fact_5102_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
        <=> ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y4))) ) ) ) ).

% mono_def
tff(fact_5103_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,F2,Y2))) )
         => pp(aa(fun(A,B),bool,order_mono(A,B),F2)) ) ) ).

% monoI
tff(fact_5104_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y))) ) ) ) ).

% monoE
tff(fact_5105_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y))) ) ) ) ).

% monoD
tff(fact_5106_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),X6))
        <=> ! [M3: nat,N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,M3)),aa(nat,A,X6,N3))) ) ) ) ).

% incseq_def
tff(fact_5107_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,I)),aa(nat,A,F2,J))) ) ) ) ).

% incseqD
tff(fact_5108_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F2))
        <=> ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F2,N3)),aa(nat,A,F2,aa(nat,nat,suc,N3)))) ) ) ).

% incseq_Suc_iff
tff(fact_5109_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,X6,aa(nat,nat,suc,N2))))
         => pp(aa(fun(nat,A),bool,order_mono(nat,A),X6)) ) ) ).

% incseq_SucI
tff(fact_5110_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I: nat] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,I)),aa(nat,A,A3,aa(nat,nat,suc,I)))) ) ) ).

% incseq_SucD
tff(fact_5111_relpow__Suc__D2_H,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A)),X2: A,Y3: A,Z4: A] :
      ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R)))
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)),R)) )
     => ? [W: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),W)),R))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R))) ) ) ).

% relpow_Suc_D2'
tff(fact_5112_filtermap__bot__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),F4: filter(B)] :
      ( ( filtermap(B,A,F2,F4) = bot_bot(filter(A)) )
    <=> ( F4 = bot_bot(filter(B)) ) ) ).

% filtermap_bot_iff
tff(fact_5113_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),Q))
         => pp(aa(fun(nat,A),bool,order_mono(nat,A),aTP_Lamp_wk(fun(A,A),fun(nat,A),Q))) ) ) ).

% mono_funpow
tff(fact_5114_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F2: fun(A,B),A3: A,B3: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A3)),aa(A,B,F2,B3)))) ) ) ).

% mono_inf
tff(fact_5115_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),A3: A,B3: A,N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),A3)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),B3))) ) ) ) ).

% funpow_mono
tff(fact_5116_relpow__Suc__I2,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Z: A,N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R))) ) ) ).

% relpow_Suc_I2
tff(fact_5117_relpow__Suc__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R)))
     => ~ ! [Y2: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R))
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R))) ) ) ).

% relpow_Suc_E2
tff(fact_5118_relpow__Suc__D2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R)))
     => ? [Y2: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R))) ) ) ).

% relpow_Suc_D2
tff(fact_5119_relpow__Suc__I,axiom,
    ! [A: $tType,X: A,Y: A,N: nat,R: set(product_prod(A,A)),Z: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),R))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R))) ) ) ).

% relpow_Suc_I
tff(fact_5120_relpow__Suc__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R)))
     => ~ ! [Y2: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R)))
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z)),R)) ) ) ).

% relpow_Suc_E
tff(fact_5121_relpow__0__I,axiom,
    ! [A: $tType,X: A,R: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R))) ).

% relpow_0_I
tff(fact_5122_relpow__0__E,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R)))
     => ( X = Y ) ) ).

% relpow_0_E
tff(fact_5123_cclfp__lowerbound,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(A,A),A3: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,A3)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),order_532582986084564980_cclfp(A,F2)),A3)) ) ) ) ).

% cclfp_lowerbound
tff(fact_5124_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aa(nat,fun(nat,nat),times_times(nat),N))) ) ).

% mono_times_nat
tff(fact_5125_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),times_times(A),A2))) ) ) ).

% mono_mult
tff(fact_5126_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F2: fun(A,B),M: A,N: A,M7: B,N6: B] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( ( aa(set(A),set(B),image2(A,B,F2),set_or7035219750837199246ssThan(A,M,N)) = set_or7035219750837199246ssThan(B,M7,N6) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
             => ( aa(A,B,F2,M) = M7 ) ) ) ) ) ).

% mono_image_least
tff(fact_5127_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F2: fun(A,A),P2: A,K: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,F2,P2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A)))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_5128_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F2: fun(A,A),P2: A,K: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,P2)),P2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A))),P2)) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_5129_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),I: nat,J: nat,X: A,Y: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F2,X)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I),F2),X)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J),F2),Y))) ) ) ) ) ) ).

% funpow_mono2
tff(fact_5130_relpowp__relpow__eq,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A)),X2: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),aTP_Lamp_wl(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X2),Xa))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),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))),N),R))) ) ).

% relpowp_relpow_eq
tff(fact_5131_filtermap__nhds__times,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C2),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ) ).

% filtermap_nhds_times
tff(fact_5132_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: fun(C,A),I6: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wm(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I6))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A3),I6))))) ) ) ).

% mono_SUP
tff(fact_5133_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3)))) ) ) ).

% mono_Sup
tff(fact_5134_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F2: fun(A,B),A3: fun(C,A),I6: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A3),I6)))),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_wm(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I6)))) ) ) ).

% mono_INF
tff(fact_5135_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3)))) ) ) ).

% mono_Inf
tff(fact_5136_relpow__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y2: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M5),R)))
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z)),R)) ) ) ) ) ).

% relpow_E
tff(fact_5137_relpow__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y2: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R))
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M5),R))) ) ) ) ) ).

% relpow_E2
tff(fact_5138_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),Q))
         => order_antimono(nat,A,aTP_Lamp_wn(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_5139_relpow__empty,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_5140_incseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L5: A,N: nat] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),X6))
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N)),L5)) ) ) ) ).

% incseq_le
tff(fact_5141_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M: nat,N: nat,F2: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F2),top_top(A)))) ) ) ) ).

% funpow_increasing
tff(fact_5142_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M: nat,N: nat,F2: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F2),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),bot_bot(A)))) ) ) ) ).

% funpow_decreasing
tff(fact_5143_mono__Max__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( pp(aa(set(A),bool,finite_finite2(A),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(A,B,F2,aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_5144_at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : topolo174197925503356063within(A,A2,top_top(set(A))) = filtermap(A,A,aTP_Lamp_wo(A,fun(A,A),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% at_to_0
tff(fact_5145_relpow__fun__conv,axiom,
    ! [A: $tType,A2: A,B2: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R)))
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = A2 )
          & ( aa(nat,A,F7,N) = B2 )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
             => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(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_5146_filtermap__times__pos__at__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [C2: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C2),topolo174197925503356063within(A,P2,aa(A,set(A),set_ord_greaterThan(A),P2))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2),aa(A,set(A),set_ord_greaterThan(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2))) ) ) ) ).

% filtermap_times_pos_at_right
tff(fact_5147_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( condit941137186595557371_above(A,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3)))) ) ) ) ) ).

% mono_cSup
tff(fact_5148_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A3: fun(C,A),I6: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,A3),I6))
           => ( ( I6 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wp(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I6))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A3),I6))))) ) ) ) ) ).

% mono_cSUP
tff(fact_5149_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( condit1013018076250108175_below(A,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3)))) ) ) ) ) ).

% mono_cInf
tff(fact_5150_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F2: fun(A,B),A3: fun(C,A),I6: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,A3),I6))
           => ( ( I6 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A3),I6)))),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_wp(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I6)))) ) ) ) ) ).

% mono_cINF
tff(fact_5151_at__to__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ( topolo174197925503356063within(A,zero_zero(A),top_top(set(A))) = filtermap(A,A,inverse_inverse(A),at_infinity(A)) ) ) ).

% at_to_infinity
tff(fact_5152_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,X: A,F4: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))),F4) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),F4) ).

% prod_filter_principal_singleton
tff(fact_5153_relpow__finite__bounded,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_wh(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_wq(set(product_prod(A,A)),fun(nat,bool),R)))))) ) ).

% relpow_finite_bounded
tff(fact_5154_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),X: B] : prod_filter(A,B,F4,principal(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_jn(B,fun(A,product_prod(A,B))),X),F4) ).

% prod_filter_principal_singleton2
tff(fact_5155_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aTP_Lamp_wr(nat,fun(nat,nat),K))) ) ).

% mono_ge2_power_minus_self
tff(fact_5156_cauchy__filter__metric__filtermap,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V768167426530841204y_dist(B)
        & topolo7287701948861334536_space(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( topolo6773858410816713723filter(B,filtermap(A,B,F2,F4))
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [P5: fun(A,bool)] :
                  ( eventually(A,P5,F4)
                  & ! [X3: A,Y4: A] :
                      ( ( pp(aa(A,bool,P5,X3))
                        & pp(aa(A,bool,P5,Y4)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X3),aa(A,B,F2,Y4))),E4)) ) ) ) ) ) ).

% cauchy_filter_metric_filtermap
tff(fact_5157_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat)))))
         => ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F2))
           => ( ! [N2: nat] :
                  ( ( aa(nat,A,F2,N2) = aa(nat,A,F2,aa(nat,nat,suc,N2)) )
                 => ( aa(nat,A,F2,aa(nat,nat,suc,N2)) = aa(nat,A,F2,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) )
             => ? [N9: nat] :
                  ( ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N5),N9))
                     => ! [M2: nat] :
                          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N9))
                         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N5))
                           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,M2)),aa(nat,A,F2,N5))) ) ) )
                  & ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N5))
                     => ( aa(nat,A,F2,N9) = aa(nat,A,F2,N5) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_5158_tendsto__at__left__sequentially,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(A) )
     => ! [B2: B,A2: B,X6: fun(B,A),L5: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),A2))
         => ( ! [S6: fun(nat,B)] :
                ( ! [N5: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(nat,B,S6,N5)),A2))
               => ( ! [N5: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),aa(nat,B,S6,N5)))
                 => ( pp(aa(fun(nat,B),bool,order_mono(nat,B),S6))
                   => ( filterlim(nat,B,S6,topolo7230453075368039082e_nhds(B,A2),at_top(nat))
                     => filterlim(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_ws(fun(B,A),fun(fun(nat,B),fun(nat,A)),X6),S6),topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ) )
           => filterlim(B,A,X6,topolo7230453075368039082e_nhds(A,L5),topolo174197925503356063within(B,A2,set_ord_lessThan(B,A2))) ) ) ) ).

% tendsto_at_left_sequentially
tff(fact_5159_continuous__at__Sup__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Sup_Sup(A),S2),set_ord_lessThan(A,aa(set(A),A,complete_Sup_Sup(A),S2))),F2)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S2)
               => ( aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),S2)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),S2)) ) ) ) ) ) ) ).

% continuous_at_Sup_mono
tff(fact_5160_continuous__at__Inf__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Inf_Inf(A),S2),aa(A,set(A),set_ord_greaterThan(A),aa(set(A),A,complete_Inf_Inf(A),S2))),F2)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S2)
               => ( aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),S2)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),S2)) ) ) ) ) ) ) ).

% continuous_at_Inf_mono
tff(fact_5161_ntrancl__def,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,N,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_wh(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_wt(nat,fun(nat,bool),N)))) ).

% ntrancl_def
tff(fact_5162_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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_wh(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_wi(set(product_prod(A,A)),fun(nat,bool),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_5163_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F7: fun(nat,nat)] :
          ( pp(aa(fun(nat,nat),bool,order_mono(nat,nat),F7))
          & ( aa(set(nat),set(nat),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)),Ys)) )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),aa(nat,nat,F7,I4)) ) )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_5164_remdups__adj__Nil__iff,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( remdups_adj(A,Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% remdups_adj_Nil_iff
tff(fact_5165_remdups__adj__set,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),remdups_adj(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% remdups_adj_set
tff(fact_5166_trancl__empty,axiom,
    ! [A: $tType] : transitive_trancl(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ).

% trancl_empty
tff(fact_5167_finite__trancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),transitive_trancl(A,R2)))
    <=> pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2)) ) ).

% finite_trancl
tff(fact_5168_ntrancl__Zero,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : transitive_ntrancl(A,zero_zero(nat),R) = R ).

% ntrancl_Zero
tff(fact_5169_converse__trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R2)))
     => ( ! [Y2: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B2)),R2))
           => pp(aa(A,bool,P,Y2)) )
       => ( ! [Y2: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_trancl(A,R2)))
               => ( pp(aa(A,bool,P,Z3))
                 => pp(aa(A,bool,P,Y2)) ) ) )
         => pp(aa(A,bool,P,A2)) ) ) ) ).

% converse_trancl_induct
tff(fact_5170_trancl__trans__induct,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),P: fun(A,fun(A,bool))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2)))
     => ( ! [X4: A,Y2: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y2)),R2))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Y2)) )
       => ( ! [X4: A,Y2: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y2)),transitive_trancl(A,R2)))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Y2))
               => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),transitive_trancl(A,R2)))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),P,Y2),Z3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Z3)) ) ) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y)) ) ) ) ).

% trancl_trans_induct
tff(fact_5171_trancl__into__trancl2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_trancl(A,R2)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R2))) ) ) ).

% trancl_into_trancl2
tff(fact_5172_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R2))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R2))) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_5173_irrefl__trancl__rD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Y: A] :
      ( ! [X4: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),transitive_trancl(A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_5174_converse__tranclE,axiom,
    ! [A: $tType,X: A,Z: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_trancl(A,R2)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),R2))
       => ~ ! [Y2: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R2))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z)),transitive_trancl(A,R2))) ) ) ) ).

% converse_tranclE
tff(fact_5175_r__r__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R))) ) ) ).

% r_r_into_trancl
tff(fact_5176_trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R2)))
     => ( ! [Y2: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y2)),R2))
           => pp(aa(A,bool,P,Y2)) )
       => ( ! [Y2: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y2)),transitive_trancl(A,R2)))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2))
               => ( pp(aa(A,bool,P,Y2))
                 => pp(aa(A,bool,P,Z3)) ) ) )
         => pp(aa(A,bool,P,B2)) ) ) ) ).

% trancl_induct
tff(fact_5177_trancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),transitive_trancl(A,R2)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_trancl(A,R2))) ) ) ).

% trancl_trans
tff(fact_5178_tranclE,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R2)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
       => ~ ! [C4: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C4)),transitive_trancl(A,R2)))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C4),B2)),R2)) ) ) ) ).

% tranclE
tff(fact_5179_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R2))) ) ).

% trancl.r_into_trancl
tff(fact_5180_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R2)))
    <=> ( ? [A7: A,B7: A] :
            ( ( A1 = A7 )
            & ( A22 = B7 )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7)),R2)) )
        | ? [A7: A,B7: A,C5: A] :
            ( ( A1 = A7 )
            & ( A22 = C5 )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7)),transitive_trancl(A,R2)))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C5)),R2)) ) ) ) ).

% trancl.simps
tff(fact_5181_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R2)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),R2))
       => ~ ! [B4: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4)),transitive_trancl(A,R2)))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22)),R2)) ) ) ) ).

% trancl.cases
tff(fact_5182_mono__Int,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A3: set(A),B3: set(A)] :
      ( pp(aa(fun(set(A),set(B)),bool,order_mono(set(A),set(B)),F2))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F2,A3)),aa(set(A),set(B),F2,B3)))) ) ).

% mono_Int
tff(fact_5183_remdups__adj__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups_adj(A,Xs) = Xs ) ) ).

% remdups_adj_distinct
tff(fact_5184_remdups__adj_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups_adj(A,nil(A)) = nil(A) ).

% remdups_adj.simps(1)
tff(fact_5185_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_trancl(product_prod(A,B),R2)))
     => ( ! [A4: A,B4: B] :
            ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))),R2))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B4)) )
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))),transitive_trancl(product_prod(A,B),R2)))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R2))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B4))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).

% trancl_induct2
tff(fact_5186_remdups__adj__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% remdups_adj_length
tff(fact_5187_filtermap__sequentually__ne__bot,axiom,
    ! [A: $tType,F2: fun(nat,A)] : filtermap(nat,A,F2,at_top(nat)) != bot_bot(filter(A)) ).

% filtermap_sequentually_ne_bot
tff(fact_5188_finite__trancl__ntranl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R))
     => ( transitive_trancl(A,R) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),R)),one_one(nat)),R) ) ) ).

% finite_trancl_ntranl
tff(fact_5189_trancl__set__ntrancl,axiom,
    ! [A: $tType,Xs: list(product_prod(A,A))] : transitive_trancl(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs))),one_one(nat)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) ).

% trancl_set_ntrancl
tff(fact_5190_trancl__power,axiom,
    ! [A: $tType,P2: product_prod(A,A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P2),transitive_trancl(A,R)))
    <=> ? [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N3),R))) ) ) ).

% trancl_power
tff(fact_5191_remdups__adj__adjacent,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I)) ) ) ).

% remdups_adj_adjacent
tff(fact_5192_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))) ) ).

% remdups_adj_length_ge1
tff(fact_5193_compact__imp__fip__image,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),I6: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I6))
               => topolo7761053866217962861closed(A,aa(B,set(A),F2,I2)) )
           => ( ! [I8: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite2(B),I8))
                 => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),I8),I6))
                   => ( 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)),aa(set(B),set(set(A)),image2(B,set(A),F2),I8))) != 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)),aa(set(B),set(set(A)),image2(B,set(A),F2),I6))) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip_image
tff(fact_5194_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => inj_on(real,real,aTP_Lamp_mn(nat,fun(real,real),N),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_5195_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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_wh(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_wq(set(product_prod(A,A)),fun(nat,bool),R)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_5196_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_5197_closed__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo7761053866217962861closed(A,bot_bot(set(A))) ) ).

% closed_empty
tff(fact_5198_closed__singleton,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [A2: A] : topolo7761053866217962861closed(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ).

% closed_singleton
tff(fact_5199_closed__Union,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(set(A))] :
          ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),S2))
         => ( ! [X4: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),S2))
               => topolo7761053866217962861closed(A,X4) )
           => topolo7761053866217962861closed(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S2)) ) ) ) ).

% closed_Union
tff(fact_5200_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_5201_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_wu(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_5202_closed__UN,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: set(B),B3: fun(B,set(A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => topolo7761053866217962861closed(A,aa(B,set(A),B3,X4)) )
           => topolo7761053866217962861closed(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A3))) ) ) ) ).

% closed_UN
tff(fact_5203_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: A,A3: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3))
    <=> ( inj_on(A,B,F2,A3)
        & ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))))) ) ) ).

% inj_on_insert
tff(fact_5204_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2))
             => ( aa(A,B,F2,X4) != aa(A,B,F2,Y2) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_5205_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A),D6: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( inj_on(B,A,F2,D6)
       => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aa(fun(B,A),fun(set(B),fun(B,bool)),aTP_Lamp_wv(set(A),fun(fun(B,A),fun(set(B),fun(B,bool))),A3),F2),D6)))) ) ) ).

% finite_inverse_image_gen
tff(fact_5206_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_kg(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_5207_finite__imp__closed,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => topolo7761053866217962861closed(A,S2) ) ) ).

% finite_imp_closed
tff(fact_5208_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,A3: set(A)] :
          ( ( A2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),A3) ) ) ).

% inj_on_mult
tff(fact_5209_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X4: A,Y2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
                 => ( aa(A,B,F2,X4) != aa(A,B,F2,Y2) ) ) ) )
         => ( ! [X4: A,Y2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
                    | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X4)) ) ) )
           => inj_on(A,B,F2,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_5210_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R2)))
     => ( ( A22 != A1 )
       => ~ ! [B4: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4)),transitive_rtrancl(A,R2)))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22)),R2)) ) ) ) ).

% rtrancl.cases
tff(fact_5211_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R2)))
    <=> ( ? [A7: A] :
            ( ( A1 = A7 )
            & ( A22 = A7 ) )
        | ? [A7: A,B7: A,C5: A] :
            ( ( A1 = A7 )
            & ( A22 = C5 )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7)),transitive_rtrancl(A,R2)))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C5)),R2)) ) ) ) ).

% rtrancl.simps
tff(fact_5212_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A2: A,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2)),transitive_rtrancl(A,R2))) ).

% rtrancl.rtrancl_refl
tff(fact_5213_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R2))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_rtrancl(A,R2))) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_5214_rtranclE,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R2)))
     => ( ( A2 != B2 )
       => ~ ! [Y2: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y2)),transitive_rtrancl(A,R2)))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),B2)),R2)) ) ) ) ).

% rtranclE
tff(fact_5215_rtrancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),transitive_rtrancl(A,R2)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_rtrancl(A,R2))) ) ) ).

% rtrancl_trans
tff(fact_5216_rtrancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R2)))
     => ( pp(aa(A,bool,P,A2))
       => ( ! [Y2: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y2)),transitive_rtrancl(A,R2)))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2))
               => ( pp(aa(A,bool,P,Y2))
                 => pp(aa(A,bool,P,Z3)) ) ) )
         => pp(aa(A,bool,P,B2)) ) ) ) ).

% rtrancl_induct
tff(fact_5217_converse__rtranclE,axiom,
    ! [A: $tType,X: A,Z: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_rtrancl(A,R2)))
     => ( ( X != Z )
       => ~ ! [Y2: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R2))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z)),transitive_rtrancl(A,R2))) ) ) ) ).

% converse_rtranclE
tff(fact_5218_converse__rtrancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R2)))
     => ( pp(aa(A,bool,P,B2))
       => ( ! [Y2: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_rtrancl(A,R2)))
               => ( pp(aa(A,bool,P,Z3))
                 => pp(aa(A,bool,P,Y2)) ) ) )
         => pp(aa(A,bool,P,A2)) ) ) ) ).

% converse_rtrancl_induct
tff(fact_5219_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R2)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_rtrancl(A,R2))) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_5220_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => inj_on(A,B,F2,B3) ) ) ).

% inj_on_subset
tff(fact_5221_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(A),A3: set(A)] :
      ( inj_on(A,B,F2,B3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => inj_on(A,B,F2,A3) ) ) ).

% subset_inj_on
tff(fact_5222_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),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_5223_inj__on__image__Fpow,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => inj_on(set(A),set(B),image2(A,B,F2),finite_Fpow(A,A3)) ) ).

% inj_on_image_Fpow
tff(fact_5224_finite__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3)))
     => ( inj_on(B,A,F2,A3)
       => pp(aa(set(B),bool,finite_finite2(B),A3)) ) ) ).

% finite_imageD
tff(fact_5225_finite__image__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),image2(A,B,F2),A3)))
      <=> pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% finite_image_iff
tff(fact_5226_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C3: set(A),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,C3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C3))
         => ( ( aa(set(A),set(B),image2(A,B,F2),A3) = aa(set(A),set(B),image2(A,B,F2),B3) )
          <=> ( A3 = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_5227_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(A),A2: A,A3: set(A)] :
      ( inj_on(A,B,F2,B3)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),aa(set(A),set(B),image2(A,B,F2),A3)))
          <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_5228_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(B,A),T3: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(B),set(A),image2(B,A,F2),T3)))
    <=> ? [U4: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),U4),T3))
          & inj_on(B,A,F2,U4)
          & ( S2 = aa(set(B),set(A),image2(B,A,F2),U4) ) ) ) ).

% subset_image_inj
tff(fact_5229_card__image,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A3)) = aa(set(A),nat,finite_card(A),A3) ) ) ).

% card_image
tff(fact_5230_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(A),A3: set(A)] :
      ( inj_on(A,B,F2,B3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B3))) ) ) ).

% inj_on_strict_subset
tff(fact_5231_inj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),N: nat] :
      ( inj_on(A,A,F2,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),top_top(set(A))) ) ).

% inj_fn
tff(fact_5232_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R2)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R2))) ) ) ).

% trancl_rtrancl_trancl
tff(fact_5233_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),transitive_trancl(A,R2)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_trancl(A,R2))) ) ) ).

% rtrancl_trancl_trancl
tff(fact_5234_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R2)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R2))) ) ) ).

% rtrancl_into_trancl2
tff(fact_5235_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R2))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R2))) ) ) ).

% rtrancl_into_trancl1
tff(fact_5236_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R)))
    <=> ( ( X = Y )
        | ( ( X != Y )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R))) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_5237_trancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R2)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R2))) ) ).

% trancl_into_rtrancl
tff(fact_5238_tranclD2,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R)))
     => ? [Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),transitive_rtrancl(A,R)))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y)),R)) ) ) ).

% tranclD2
tff(fact_5239_rtranclD,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R)))
     => ( ( A2 = B2 )
        | ( ( A2 != B2 )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R))) ) ) ) ).

% rtranclD
tff(fact_5240_tranclD,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R)))
     => ? [Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),R))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y)),transitive_rtrancl(A,R))) ) ) ).

% tranclD
tff(fact_5241_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R2)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By))
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R2))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R2)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B4)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay)) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_5242_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xa2: A,Xb2: B,Za: A,Zb: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R2)))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb2) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
       => ~ ! [A4: A,B4: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))),R2))
             => ~ pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R2))) ) ) ) ).

% converse_rtranclE2
tff(fact_5243_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R2)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay))
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))),transitive_rtrancl(product_prod(A,B),R2)))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R2))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B4))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).

% rtrancl_induct2
tff(fact_5244_inj__on__Inter,axiom,
    ! [B: $tType,A: $tType,S2: set(set(A)),F2: fun(A,B)] :
      ( ( S2 != bot_bot(set(set(A))) )
     => ( ! [A5: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A5),S2))
           => inj_on(A,B,F2,A5) )
       => inj_on(A,B,F2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S2)) ) ) ).

% inj_on_Inter
tff(fact_5245_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( inj_on(B,A,F2,top_top(set(B)))
       => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ww(set(A),fun(fun(B,A),fun(B,bool)),A3),F2)))) ) ) ).

% finite_inverse_image
tff(fact_5246_finite__UNIV__inj__surj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( pp(aa(set(A),bool,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_5247_finite__UNIV__surj__inj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( pp(aa(set(A),bool,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_5248_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B3)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ).

% inj_image_subset_iff
tff(fact_5249_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),A11: set(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ? [F7: fun(A,B)] :
            ( inj_on(A,B,F7,A3)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A3)),A11)) )
      <=> ? [G7: fun(B,A)] : aa(set(B),set(A),image2(B,A,G7),A11) = A3 ) ) ).

% inj_on_iff_surj
tff(fact_5250_finite__surj__inj,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),image2(A,A,F2),A3)))
       => inj_on(A,A,F2,A3) ) ) ).

% finite_surj_inj
tff(fact_5251_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B3: set(B)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3))
       => ( pp(aa(set(B),bool,finite_finite2(B),B3))
         => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ) ).

% inj_on_finite
tff(fact_5252_endo__inj__surj,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image2(A,A,F2),A3)),A3))
       => ( inj_on(A,A,F2,A3)
         => ( aa(set(A),set(A),image2(A,A,F2),A3) = A3 ) ) ) ) ).

% endo_inj_surj
tff(fact_5253_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C3: set(A),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,C3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C3))
         => ( 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)),A3),B3)) = 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),A3)),aa(set(A),set(B),image2(A,B,F2),B3)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_5254_eq__card__imp__inj__on,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A3)) = aa(set(A),nat,finite_card(A),A3) )
       => inj_on(A,B,F2,A3) ) ) ).

% eq_card_imp_inj_on
tff(fact_5255_inj__on__iff__eq__card,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( inj_on(A,B,F2,A3)
      <=> ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A3)) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_5256_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C3: set(A),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,C3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),C3))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C3))
         => ( aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B3)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_5257_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(B),nat,finite_card(B),A3)))
     => ~ inj_on(B,A,F2,A3) ) ).

% pigeonhole
tff(fact_5258_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: A,X: A,B2: A,F2: fun(A,B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
             => ( inj_on(A,B,F2,set_or1337092689740270186AtMost(A,A2,B2))
               => ( ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,A2)),aa(A,B,F2,X)))
                    & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,B2))) )
                  | ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,B2)),aa(A,B,F2,X)))
                    & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,A2))) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_5259_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),X: B,B3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(set(A),set(B),image2(A,B,F2),A3)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),the_inv_into(A,B,A3,F2,X)),B3)) ) ) ) ).

% the_inv_into_into
tff(fact_5260_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),transitive_rtrancl(list(A),listrel1(A,R2))))
     => ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_5261_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I6: set(A),A3: fun(A,set(B)),F2: fun(B,C)] :
      ( ! [I2: A,J2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I6))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,I2)),aa(A,set(B),A3,J2)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I2))) ) ) )
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
           => inj_on(B,C,F2,aa(A,set(B),A3,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),A3),I6))) ) ) ).

% inj_on_UNION_chain
tff(fact_5262_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I6: set(A),F2: fun(B,C),A3: fun(A,set(B))] :
      ( ( I6 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
           => inj_on(B,C,F2,aa(A,set(B),A3,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),A3),I6))) ) ) ).

% inj_on_INTER
tff(fact_5263_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,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_wx(fun(A,B),fun(fun(A,B),fun(A,bool)),F2),G))) ) ) ) ).

% closed_Collect_le
tff(fact_5264_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3))
       => ( inj_on(B,A,G,B3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),B3)),A3))
           => ( pp(aa(set(A),bool,finite_finite2(A),A3))
             => ( pp(aa(set(B),bool,finite_finite2(B),B3))
               => ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_5265_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S2: set(A),T3: set(B),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => ( pp(aa(set(B),bool,finite_finite2(B),T3))
       => ( ( aa(set(A),nat,finite_card(A),S2) = aa(set(B),nat,finite_card(B),T3) )
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),S2)),T3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),T3))
                 => ? [Xa3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),S2))
                      & ( aa(A,B,F2,Xa3) = X3 ) ) )
            <=> inj_on(A,B,F2,S2) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_5266_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),uminus_uminus(set(A)),A3))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)))) ) ).

% inj_image_Compl_subset
tff(fact_5267_t3__space,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [S2: set(A),Y: A] :
          ( topolo7761053866217962861closed(A,S2)
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
           => ? [U5: set(A),V5: set(A)] :
                ( topolo1002775350975398744n_open(A,U5)
                & topolo1002775350975398744n_open(A,V5)
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),U5))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),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_5268_t4__space,axiom,
    ! [A: $tType] :
      ( topological_t4_space(A)
     => ! [S2: set(A),T3: set(A)] :
          ( topolo7761053866217962861closed(A,S2)
         => ( topolo7761053866217962861closed(A,T3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3) = bot_bot(set(A)) )
             => ? [U5: set(A),V5: set(A)] :
                  ( topolo1002775350975398744n_open(A,U5)
                  & topolo1002775350975398744n_open(A,V5)
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),U5))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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_5269_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),C3: set(A),A3: set(C),B3: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F2,C3)
     => ( ! [X4: C] :
            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X4),A3))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(C,set(A),B3,X4)),C3)) )
       => ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J),A3))
         => ( 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),B3),A3))) = 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_wy(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F2),B3)),A3)) ) ) ) ) ).

% image_INT
tff(fact_5270_nhds__closed,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ( topolo1002775350975398744n_open(A,A3)
           => ? [A14: set(A)] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A14))
                & topolo7761053866217962861closed(A,A14)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A14),A3))
                & eventually(A,aTP_Lamp_wz(set(A),fun(A,bool),A14),topolo7230453075368039082e_nhds(A,X)) ) ) ) ) ).

% nhds_closed
tff(fact_5271_continuous__on__closed__Union,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [I6: set(A),U2: fun(A,set(B)),F2: fun(B,C)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I6))
         => ( ! [I2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
               => topolo7761053866217962861closed(B,aa(A,set(B),U2,I2)) )
           => ( ! [I2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
                 => 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),I6)),F2) ) ) ) ) ).

% continuous_on_closed_Union
tff(fact_5272_Lim__in__closed__set,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F2: fun(B,A),F4: filter(B),L: A] :
          ( topolo7761053866217962861closed(A,S2)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_xa(set(A),fun(fun(B,A),fun(B,bool)),S2),F2),F4)
           => ( ( F4 != bot_bot(filter(B)) )
             => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
               => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),L),S2)) ) ) ) ) ) ).

% Lim_in_closed_set
tff(fact_5273_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3)))
         => ? [F3: fun(A,B)] :
              ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A3)),B3))
              & inj_on(A,B,F3,A3) ) ) ) ) ).

% card_le_inj
tff(fact_5274_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3))
       => ( pp(aa(set(B),bool,finite_finite2(B),B3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3))) ) ) ) ).

% card_inj_on_le
tff(fact_5275_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B3))
       => ( ? [F7: fun(A,B)] :
              ( inj_on(A,B,F7,A3)
              & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A3)),B3)) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3))) ) ) ) ).

% inj_on_iff_card_le
tff(fact_5276_log__inj,axiom,
    ! [B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => inj_on(real,real,log(B2),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))) ) ).

% log_inj
tff(fact_5277_compact__imp__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F4: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ! [T6: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),T6),F4))
               => topolo7761053866217962861closed(A,T6) )
           => ( ! [F11: set(set(A))] :
                  ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),F11))
                 => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F11),F4))
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),F11)) != bot_bot(set(A)) ) ) )
             => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),F4)) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip
tff(fact_5278_compact__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U2: set(A)] :
          ( topolo2193935891317330818ompact(A,U2)
        <=> ! [A8: set(set(A))] :
              ( ! [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A8))
                 => topolo7761053866217962861closed(A,X3) )
             => ( ! [B10: set(set(A))] :
                    ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),B10),A8))
                   => ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),B10))
                     => ( 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)),B10)) != 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_5279_ex__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ? [T10: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T10),aa(set(B),set(A),image2(B,A,F2),S2)))
          & pp(aa(set(A),bool,P,T10)) )
    <=> ? [T10: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T10),S2))
          & inj_on(B,A,F2,T10)
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F2),T10))) ) ) ).

% ex_subset_image_inj
tff(fact_5280_all__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ! [T10: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T10),aa(set(B),set(A),image2(B,A,F2),S2)))
         => pp(aa(set(A),bool,P,T10)) )
    <=> ! [T10: set(B)] :
          ( ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T10),S2))
            & inj_on(B,A,F2,T10) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F2),T10))) ) ) ).

% all_subset_image_inj
tff(fact_5281_uniformly__continuous__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S: set(A),F2: fun(A,B),E5: fun(product_prod(B,B),bool)] :
          ( topolo6026614971017936543ous_on(A,B,S,F2)
         => ( eventually(product_prod(B,B),E5,topolo7806501430040627800ormity(B))
           => eventually(product_prod(A,A),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(product_prod(B,B),bool),fun(A,fun(A,bool)),aa(fun(A,B),fun(fun(product_prod(B,B),bool),fun(A,fun(A,bool))),aTP_Lamp_xb(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),bool),fun(A,fun(A,bool)))),S),F2),E5)),topolo7806501430040627800ormity(A)) ) ) ) ).

% uniformly_continuous_onD
tff(fact_5282_inj__Some,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,option(A),some(A),A3) ).

% inj_Some
tff(fact_5283_inj__Suc,axiom,
    ! [N4: set(nat)] : inj_on(nat,nat,suc,N4) ).

% inj_Suc
tff(fact_5284_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X6: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_xc(fun(A,B),fun(A,product_prod(A,B)),F2),X6) ).

% inj_on_convol_ident
tff(fact_5285_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_5286_inj__singleton,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,set(A),aTP_Lamp_lp(A,set(A)),A3) ).

% inj_singleton
tff(fact_5287_inj__on__diff__nat,axiom,
    ! [N4: set(nat),K: nat] :
      ( ! [N2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2)) )
     => inj_on(nat,nat,aTP_Lamp_kh(nat,fun(nat,nat),K),N4) ) ).

% inj_on_diff_nat
tff(fact_5288_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A3: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_jo(A,fun(B,product_prod(B,A)))),A3) ).

% swap_inj_on
tff(fact_5289_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,aa(fun(set(nat),bool),set(set(nat)),collect(set(nat)),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_5290_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_xe(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_5291_range__inj__infinite,axiom,
    ! [A: $tType,F2: fun(nat,A)] :
      ( inj_on(nat,A,F2,top_top(set(nat)))
     => ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat))))) ) ).

% range_inj_infinite
tff(fact_5292_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [N2: nat,F3: fun(nat,A)] :
          ( ( A3 = aa(set(nat),set(A),image2(nat,A,F3),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N2))) )
          & inj_on(nat,A,F3,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N2))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_5293_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [F3: fun(A,nat),N2: nat] :
          ( ( aa(set(A),set(nat),image2(A,nat,F3),A3) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N2)) )
          & inj_on(A,nat,F3,A3) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_5294_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X4: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),I6))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),aa(list(A),nat,size_size(list(A)),Xs))) )
       => inj_on(nat,A,nth(A,Xs),I6) ) ) ).

% inj_on_nth
tff(fact_5295_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
    <=> ? [F7: fun(nat,A)] :
          ( inj_on(nat,A,F7,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F7),top_top(set(nat)))),S2)) ) ) ).

% infinite_iff_countable_subset
tff(fact_5296_infinite__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
     => ? [F3: fun(nat,A)] :
          ( inj_on(nat,A,F3,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F3),top_top(set(nat)))),S2)) ) ) ).

% infinite_countable_subset
tff(fact_5297_inj__on__funpow__least,axiom,
    ! [A: $tType,N: nat,F2: fun(A,A),S: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),S) = S )
     => ( ! [M5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M5))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M5),F2),S) != S ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_xf(fun(A,A),fun(A,fun(nat,A)),F2),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_5298_uniformly__continuous__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S,F2)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [D5: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
                  & ! [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                     => ! [Xa3: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),S))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Xa3,X3)),D5))
                           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,Xa3),aa(A,B,F2,X3))),E4)) ) ) ) ) ) ) ) ).

% uniformly_continuous_on_def
tff(fact_5299_isUCont__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,top_top(set(A)),F2)
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [S7: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S7))
                  & ! [X3: A,Y4: A] :
                      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,Y4)),S7))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X3),aa(A,B,F2,Y4))),R5)) ) ) ) ) ) ).

% isUCont_def
tff(fact_5300_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F2: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_xg(fun(B,list(A)),fun(B,set(A)),F2)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_5301_has__derivative__power__int_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,N: int,S2: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,aTP_Lamp_xh(int,fun(A,A),N),aa(int,fun(A,A),aTP_Lamp_xi(A,fun(int,fun(A,A)),X),N),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_power_int'
tff(fact_5302_has__derivative__power__int,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(C,A),X: C,F9: fun(C,A),S2: set(C),N: int] :
          ( ( aa(C,A,F2,X) != zero_zero(A) )
         => ( has_derivative(C,A,F2,F9,topolo174197925503356063within(C,X,S2))
           => has_derivative(C,A,aa(int,fun(C,A),aTP_Lamp_xj(fun(C,A),fun(int,fun(C,A)),F2),N),aa(int,fun(C,A),aa(fun(C,A),fun(int,fun(C,A)),aa(C,fun(fun(C,A),fun(int,fun(C,A))),aTP_Lamp_xk(fun(C,A),fun(C,fun(fun(C,A),fun(int,fun(C,A)))),F2),X),F9),N),topolo174197925503356063within(C,X,S2)) ) ) ) ).

% has_derivative_power_int
tff(fact_5303_bind__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,list(A))] : bind(B,A,nil(B),F2) = nil(A) ).

% bind_simps(1)
tff(fact_5304_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_5305_power__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] :
          ( ( power_int(A,X,N) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( N != zero_zero(int) ) ) ) ) ).

% power_int_eq_0_iff
tff(fact_5306_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A2,N)),power_int(A,B2,N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_5307_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_less_power_int
tff(fact_5308_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_le_power_int
tff(fact_5309_power__int__not__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( N = zero_zero(int) ) )
         => ( power_int(A,X,N) != zero_zero(A) ) ) ) ).

% power_int_not_zero
tff(fact_5310_continuous__on__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),N: int] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S))
               => ( aa(A,B,F2,X4) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aa(int,fun(A,B),aTP_Lamp_xl(fun(A,B),fun(int,fun(A,B)),F2),N)) ) ) ) ).

% continuous_on_power_int
tff(fact_5311_list__bind__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F2: fun(A,list(B)),G: fun(A,list(B))] :
      ( ( Xs = Ys )
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(A,list(B),F2,X4) = aa(A,list(B),G,X4) ) )
       => ( bind(A,B,Xs,F2) = bind(A,B,Ys,G) ) ) ) ).

% list_bind_cong
tff(fact_5312_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] :
          ( ( ( M = zero_zero(int) )
           => ( power_int(A,zero_zero(A),M) = one_one(A) ) )
          & ( ( M != zero_zero(int) )
           => ( power_int(A,zero_zero(A),M) = zero_zero(A) ) ) ) ) ).

% power_int_0_left_If
tff(fact_5313_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N4: int,A2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A2,N)),power_int(A,A2,N4))) ) ) ) ).

% power_int_increasing
tff(fact_5314_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N4: int,A2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A2,N)),power_int(A,A2,N4))) ) ) ) ).

% power_int_strict_increasing
tff(fact_5315_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,M: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != N ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),M),N)) = divide_divide(A,power_int(A,X,M),power_int(A,X,N)) ) ) ) ).

% power_int_diff
tff(fact_5316_tendsto__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(B,A),A2: A,F4: filter(B),N: int] :
          ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( A2 != zero_zero(A) )
           => filterlim(B,A,aa(int,fun(B,A),aTP_Lamp_xm(fun(B,A),fun(int,fun(B,A)),F2),N),topolo7230453075368039082e_nhds(A,power_int(A,A2,N)),F4) ) ) ) ).

% tendsto_power_int
tff(fact_5317_continuous__at__within__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,S: set(A),F2: fun(A,B),N: int] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aa(int,fun(A,B),aTP_Lamp_xn(fun(A,B),fun(int,fun(A,B)),F2),N)) ) ) ) ).

% continuous_at_within_power_int
tff(fact_5318_differentiable__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S: set(A),N: int] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,B,F2,X) != zero_zero(B) )
           => differentiable(A,B,aa(int,fun(A,B),aTP_Lamp_xo(fun(A,B),fun(int,fun(A,B)),F2),N),topolo174197925503356063within(A,X,S)) ) ) ) ).

% differentiable_power_int
tff(fact_5319_continuous__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),N: int] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_rg(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aa(int,fun(A,B),aTP_Lamp_xn(fun(A,B),fun(int,fun(A,B)),F2),N)) ) ) ) ).

% continuous_power_int
tff(fact_5320_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N4: int,A2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A2,N4)),power_int(A,A2,N))) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_5321_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),power_int(A,Y,N))) ) ) ) ) ).

% power_int_mono
tff(fact_5322_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,B2,N)),power_int(A,A2,N))) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_5323_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),power_int(A,X,N))) ) ) ) ).

% one_le_power_int
tff(fact_5324_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),power_int(A,A2,N))) ) ) ) ).

% one_less_power_int
tff(fact_5325_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N) != zero_zero(int) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,X,N)) ) ) ) ).

% power_int_add
tff(fact_5326_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,B2,N)),power_int(A,A2,N))) ) ) ) ) ).

% power_int_antimono
tff(fact_5327_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A2,N)),power_int(A,B2,N))) ) ) ) ) ).

% power_int_strict_mono
tff(fact_5328_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N4: int,A2: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
             => ( ( ( A2 != zero_zero(A) )
                  | ( N4 != zero_zero(int) )
                  | ( N = zero_zero(int) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A2,N4)),power_int(A,A2,N))) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_5329_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),one_one(A))) ) ) ) ) ).

% power_int_le_one
tff(fact_5330_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,M)),power_int(A,X,N)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N)) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_5331_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,X,M)),power_int(A,X,N)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N)) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_5332_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( N != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int)))),X) = power_int(A,X,N) ) ) ) ).

% power_int_minus_mult
tff(fact_5333_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),X) ) ) ) ).

% power_int_add_1
tff(fact_5334_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,M)) ) ) ) ).

% power_int_add_1'
tff(fact_5335_powr__real__of__int_H,axiom,
    ! [X: real,N: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( ( X != zero_zero(real) )
          | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) )
       => ( powr(real,X,aa(int,real,ring_1_of_int(real),N)) = power_int(real,X,N) ) ) ) ).

% powr_real_of_int'
tff(fact_5336_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S: set(A),N: int] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_xp(fun(A,A),fun(int,fun(A,A)),F2),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),N)),power_int(A,aa(A,A,F2,X),aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int))))),D2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_power_int
tff(fact_5337_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_xq(fun(A,B),fun(B,fun(A,bool)),F2),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_5338_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_xr(fun(A,B),fun(B,fun(A,bool)),F2),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_5339_folding__idem__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
    <=> ( finite_folding_on(A,B,S2,F2)
        & finite6916993218817215295axioms(A,B,S2,F2) ) ) ).

% folding_idem_on_def
tff(fact_5340_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_eq_0_iff
tff(fact_5341_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),archimedean_frac(A,X)))
        <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_gt_0_iff
tff(fact_5342_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),ring_1_Ints(A))) ) ).

% Ints_0
tff(fact_5343_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_5344_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_xs(A,fun(A,fun(A,bool)),A2),B2)))) ) ).

% finite_int_segment
tff(fact_5345_folding__idem__on__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),S2))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X4)),aa(A,fun(B,B),F2,X4)) = aa(A,fun(B,B),F2,X4) ) ) )
     => finite6916993218817215295axioms(A,B,S2,F2) ) ).

% folding_idem_on_axioms.intro
tff(fact_5346_folding__idem__on__axioms__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite6916993218817215295axioms(A,B,S2,F2)
    <=> ! [X3: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S2))
           => ( 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) ) ) ) ) ).

% folding_idem_on_axioms_def
tff(fact_5347_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),A2) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_5348_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_xt(A,fun(A,bool),A2)))) ) ).

% finite_abs_int_segment
tff(fact_5349_folding__idem__on_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
     => finite6916993218817215295axioms(A,B,S2,F2) ) ).

% folding_idem_on.axioms(2)
tff(fact_5350_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),A2)),zero_zero(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% Ints_odd_less_0
tff(fact_5351_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( ( X != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5352_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5353_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),ring_1_Ints(A)))
           => ( ( X = Y )
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y))),one_one(A))) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5354_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = zero_zero(A) ) )
          & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,X)) ) ) ) ) ).

% frac_neg
tff(fact_5355_le__mult__floor__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A2: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A2))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A2)),archim6421214686448440834_floor(B,B2)))),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_5356_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: A] :
          ( ( archimedean_frac(A,X) = A2 )
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2)),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A))) ) ) ) ).

% frac_unique_iff
tff(fact_5357_mult__ceiling__le__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A2: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A2))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A2)),archimedean_ceiling(B,B2))))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5358_folding__idem__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite_folding_on(A,B,S2,F2)
     => ( finite6916993218817215295axioms(A,B,S2,F2)
       => finite1890593828518410140dem_on(A,B,S2,F2) ) ) ).

% folding_idem_on.intro
tff(fact_5359_UN__le__eq__Un0,axiom,
    ! [A: $tType,M4: fun(nat,set(A)),N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M4),set_ord_atMost(nat,N))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M4),set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M4,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_5360_finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( finite5375528669736107172at_top(A,A3) = principal(set(A),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),A3),bot_bot(set(set(A))))) ) ) ).

% finite_subsets_at_top_finite
tff(fact_5361_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa2)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ( ( Deg2 = Xa2 )
                & ! [X4: vEBT_VEBT] :
                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                   => vEBT_VEBT_valid(X4,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                & vEBT_VEBT_valid(Summary3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_xu(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary3)),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5362_Un__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% Un_iff
tff(fact_5363_UnCI,axiom,
    ! [A: $tType,C2: A,B3: set(A),A3: set(A)] :
      ( ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3)) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) ) ).

% UnCI
tff(fact_5364_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% sup.bounded_iff
tff(fact_5365_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ) ).

% le_sup_iff
tff(fact_5366_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),X) = X ) ).

% sup_bot_left
tff(fact_5367_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% sup_bot_right
tff(fact_5368_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_5369_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = bot_bot(A) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_5370_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = bot_bot(A) )
        <=> ( ( A2 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_5371_sup__bot_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A2) = A2 ) ).

% sup_bot.left_neutral
tff(fact_5372_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
        <=> ( ( A2 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_5373_sup__bot_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),bot_bot(A)) = A2 ) ).

% sup_bot.right_neutral
tff(fact_5374_ball__empty,axiom,
    ! [A: $tType,P: fun(A,bool),X2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),bot_bot(set(A))))
     => pp(aa(A,bool,P,X2)) ) ).

% ball_empty
tff(fact_5375_Un__empty,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ( ( A3 = bot_bot(set(A)) )
        & ( B3 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_5376_finite__Un,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G5)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),F4))
        & pp(aa(set(A),bool,finite_finite2(A),G5)) ) ) ).

% finite_Un
tff(fact_5377_Un__subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C3)) ) ) ).

% Un_subset_iff
tff(fact_5378_Un__insert__left,axiom,
    ! [A: $tType,A2: A,B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)),C3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% Un_insert_left
tff(fact_5379_Un__insert__right,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ).

% Un_insert_right
tff(fact_5380_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S2: 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)),S2),T3)),S2) = S2 ).

% Un_Int_eq(1)
tff(fact_5381_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S2: 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)),S2),T3)),T3) = T3 ).

% Un_Int_eq(2)
tff(fact_5382_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)) = S2 ).

% Un_Int_eq(3)
tff(fact_5383_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S2: 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)),S2),T3)) = T3 ).

% Un_Int_eq(4)
tff(fact_5384_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S2: 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)),S2),T3)),S2) = S2 ).

% Int_Un_eq(1)
tff(fact_5385_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S2: 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)),S2),T3)),T3) = T3 ).

% Int_Un_eq(2)
tff(fact_5386_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)) = S2 ).

% Int_Un_eq(3)
tff(fact_5387_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S2: 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)),S2),T3)) = T3 ).

% Int_Un_eq(4)
tff(fact_5388_Un__Diff__cancel,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) ).

% Un_Diff_cancel
tff(fact_5389_Un__Diff__cancel2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3) ).

% Un_Diff_cancel2
tff(fact_5390_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_xv(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),P),Q))))
      <=> ! [Y4: A] :
            ( pp(aa(A,bool,P,Y4))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_xw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q),Y4)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_5391_ball__UNIV,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),top_top(set(A))))
         => pp(aa(A,bool,P,X3)) )
    <=> ! [X_13: A] : pp(aa(A,bool,P,X_13)) ) ).

% ball_UNIV
tff(fact_5392_Compl__Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),B3) ).

% Compl_Diff_eq
tff(fact_5393_eventually__finite__subsets__at__top__weakI,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),bool)] :
      ( ! [X7: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X7))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),A3))
           => pp(aa(set(A),bool,P,X7)) ) )
     => eventually(set(A),P,finite5375528669736107172at_top(A,A3)) ) ).

% eventually_finite_subsets_at_top_weakI
tff(fact_5394_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,bool),F2: fun(B,A),G: fun(B,A),S2: 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_xx(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F2),G)),S2) = 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)),S2),aa(fun(B,bool),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)),S2),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_xy(fun(B,bool),fun(B,bool),P))))) ).

% if_image_distrib
tff(fact_5395_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),union(A),Xs),Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_union
tff(fact_5396_UN__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C3: set(C),A3: fun(C,set(D)),B3: set(D)] :
      ( ( ( C3 = bot_bot(set(C)) )
       => ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_xz(fun(C,set(D)),fun(set(D),fun(C,set(D))),A3),B3)),C3)) = bot_bot(set(D)) ) )
      & ( ( C3 != bot_bot(set(C)) )
       => ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_xz(fun(C,set(D)),fun(set(D),fun(C,set(D))),A3),B3)),C3)) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A3),C3))),B3) ) ) ) ).

% UN_simps(2)
tff(fact_5397_UN__simps_I3_J,axiom,
    ! [E: $tType,F: $tType,C3: set(F),A3: set(E),B3: fun(F,set(E))] :
      ( ( ( C3 = bot_bot(set(F)) )
       => ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),aa(fun(F,set(E)),fun(F,set(E)),aTP_Lamp_ya(set(E),fun(fun(F,set(E)),fun(F,set(E))),A3),B3)),C3)) = bot_bot(set(E)) ) )
      & ( ( C3 != bot_bot(set(F)) )
       => ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),aa(fun(F,set(E)),fun(F,set(E)),aTP_Lamp_ya(set(E),fun(fun(F,set(E)),fun(F,set(E))),A3),B3)),C3)) = aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A3),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),B3),C3))) ) ) ) ).

% UN_simps(3)
tff(fact_5398_cSUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),B3: set(B)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( ( B3 != bot_bot(set(B)) )
             => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),B3))
               => ( aa(set(A),A,complete_Sup_Sup(A),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)),A3),B3))) = 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),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),B3))) ) ) ) ) ) ) ).

% cSUP_union
tff(fact_5399_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> pp(case_option(bool,A,fFalse,aTP_Lamp_mf(A,bool),Option)) ) ).

% option.disc_eq_case(2)
tff(fact_5400_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> pp(case_option(bool,A,fTrue,aTP_Lamp_ao(A,bool),Option)) ) ).

% option.disc_eq_case(1)
tff(fact_5401_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,bool),F2: fun(A,fun(B,C))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),Q)))
       => pp(aa(set(C),bool,finite_finite2(C),aa(fun(C,bool),set(C),collect(C),aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_yb(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),P),Q),F2)))) ) ) ).

% finite_image_set2
tff(fact_5402_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
     => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aTP_Lamp_yc(fun(A,bool),fun(fun(A,B),fun(B,bool)),P),F2)))) ) ).

% finite_image_set
tff(fact_5403_option_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,B),Option: option(A)] : aa(B,C,H,case_option(B,A,F1,F22,Option)) = case_option(C,A,aa(B,C,H,F1),aa(fun(A,B),fun(A,C),aTP_Lamp_yd(fun(B,C),fun(fun(A,B),fun(A,C)),H),F22),Option) ).

% option.case_distrib
tff(fact_5404_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ab(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(fun(A,bool),set(A),collect(A),Q)) ).

% Collect_disj_eq
tff(fact_5405_Un__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ye(set(A),fun(set(A),fun(A,bool)),A3),B3)) ).

% Un_def
tff(fact_5406_insert__def,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bp(A,fun(A,bool),A2))),B3) ).

% insert_def
tff(fact_5407_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_yf(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,bool),set(A),collect(A),P))),aa(fun(A,bool),set(A),collect(A),Q)) ).

% Collect_imp_eq
tff(fact_5408_Un__UNIV__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B3) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_5409_Un__UNIV__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_5410_Un__Int__distrib2,axiom,
    ! [A: $tType,B3: set(A),C3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A3)) ).

% Un_Int_distrib2
tff(fact_5411_Int__Un__distrib2,axiom,
    ! [A: $tType,B3: set(A),C3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)) ).

% Int_Un_distrib2
tff(fact_5412_Un__Int__distrib,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3)) ).

% Un_Int_distrib
tff(fact_5413_Int__Un__distrib,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3)) ).

% Int_Un_distrib
tff(fact_5414_Un__Int__crazy,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C3),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),A3)) ).

% Un_Int_crazy
tff(fact_5415_Un__left__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3)) ).

% Un_left_commute
tff(fact_5416_Un__left__absorb,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) ).

% Un_left_absorb
tff(fact_5417_Un__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3) ).

% Un_commute
tff(fact_5418_Un__absorb,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),A3) = A3 ).

% Un_absorb
tff(fact_5419_Un__assoc,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) ).

% Un_assoc
tff(fact_5420_ball__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))
         => pp(aa(A,bool,P,X3)) )
    <=> ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(A,bool,P,X3)) )
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3))
           => pp(aa(A,bool,P,X3)) ) ) ) ).

% ball_Un
tff(fact_5421_bex__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,bool)] :
      ( ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))
          & pp(aa(A,bool,P,X3)) )
    <=> ( ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
            & pp(aa(A,bool,P,X3)) )
        | ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3))
            & pp(aa(A,bool,P,X3)) ) ) ) ).

% bex_Un
tff(fact_5422_UnI2,axiom,
    ! [A: $tType,C2: A,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) ) ).

% UnI2
tff(fact_5423_UnI1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) ) ).

% UnI1
tff(fact_5424_UnE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% UnE
tff(fact_5425_Un__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C3)) ).

% Un_Diff
tff(fact_5426_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% less_supI1
tff(fact_5427_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% less_supI2
tff(fact_5428_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb3
tff(fact_5429_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_5430_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% sup.strict_boundedE
tff(fact_5431_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_5432_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% sup.strict_coboundedI1
tff(fact_5433_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% sup.strict_coboundedI2
tff(fact_5434_finite__UnI,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,finite_finite2(A),G5))
       => pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G5))) ) ) ).

% finite_UnI
tff(fact_5435_Un__infinite,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
     => ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3))) ) ).

% Un_infinite
tff(fact_5436_infinite__Un,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)))
    <=> ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
        | ~ pp(aa(set(A),bool,finite_finite2(A),T3)) ) ) ).

% infinite_Un
tff(fact_5437_option_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F22: fun(A,B)] : case_option(B,A,F1,F22,none(A)) = F1 ).

% option.simps(4)
tff(fact_5438_option_Osimps_I5_J,axiom,
    ! [B: $tType,A: $tType,F1: B,F22: fun(A,B),X22: A] : case_option(B,A,F1,F22,aa(A,option(A),some(A),X22)) = aa(A,B,F22,X22) ).

% option.simps(5)
tff(fact_5439_finite__subsets__at__top__neq__bot,axiom,
    ! [A: $tType,A3: set(A)] : finite5375528669736107172at_top(A,A3) != bot_bot(filter(set(A))) ).

% finite_subsets_at_top_neq_bot
tff(fact_5440_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_5441_Un__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),bot_bot(set(A))) = A3 ).

% Un_empty_right
tff(fact_5442_Un__empty__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B3) = B3 ).

% Un_empty_left
tff(fact_5443_cSup__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A3)
           => ( ( B3 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,B3)
               => ( 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)),A3),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_5444_open__diagonal__complement,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_yg(product_prod(A,A),bool))) ) ).

% open_diagonal_complement
tff(fact_5445_Un__Pow__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow(A,A3)),pow(A,B3))),pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))) ).

% Un_Pow_subset
tff(fact_5446_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% sup.coboundedI2
tff(fact_5447_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% sup.coboundedI1
tff(fact_5448_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_5449_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb_iff1
tff(fact_5450_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ).

% sup.cobounded2
tff(fact_5451_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ).

% sup.cobounded1
tff(fact_5452_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.order_iff
tff(fact_5453_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)) ) ) ) ).

% sup.boundedI
tff(fact_5454_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% sup.boundedE
tff(fact_5455_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_5456_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).

% sup_absorb1
tff(fact_5457_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_5458_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb1
tff(fact_5459_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X4: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),aa(A,A,aa(A,fun(A,A),F2,X4),Y2)))
         => ( ! [X4: A,Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),aa(A,A,aa(A,fun(A,A),F2,X4),Y2)))
           => ( ! [X4: A,Y2: A,Z3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),X4))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),X4))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y2),Z3)),X4)) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F2,X),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_5460_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% sup.orderI
tff(fact_5461_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.orderE
tff(fact_5462_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_5463_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),X)) ) ) ) ).

% sup_least
tff(fact_5464_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2))) ) ) ) ).

% sup_mono
tff(fact_5465_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ) ).

% sup.mono
tff(fact_5466_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% le_supI2
tff(fact_5467_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% le_supI1
tff(fact_5468_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge2
tff(fact_5469_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge1
tff(fact_5470_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),X)) ) ) ) ).

% le_supI
tff(fact_5471_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),X))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X)) ) ) ) ).

% le_supE
tff(fact_5472_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(3)
tff(fact_5473_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(4)
tff(fact_5474_Un__mono,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A),D6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),D6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),D6))) ) ) ).

% Un_mono
tff(fact_5475_Un__least,axiom,
    ! [A: $tType,A3: set(A),C3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3)) ) ) ).

% Un_least
tff(fact_5476_Un__upper1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) ).

% Un_upper1
tff(fact_5477_Un__upper2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) ).

% Un_upper2
tff(fact_5478_Un__absorb1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_5479_Un__absorb2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = A3 ) ) ).

% Un_absorb2
tff(fact_5480_subset__UnE,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))
     => ~ ! [A14: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A14),A3))
           => ! [B12: set(A)] :
                ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B12),B3))
               => ( C3 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A14),B12) ) ) ) ) ).

% subset_UnE
tff(fact_5481_subset__Un__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_5482_Ball__Collect,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(A,bool,P,X3)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(fun(A,bool),set(A),collect(A),P))) ) ).

% Ball_Collect
tff(fact_5483_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_yh(fun(B,A),fun(fun(B,bool),fun(A,bool)),F2),P)) = aa(set(B),set(A),image2(B,A,F2),aa(fun(B,bool),set(B),collect(B),P)) ).

% setcompr_eq_image
tff(fact_5484_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_yi(fun(B,A),fun(set(B),fun(A,bool)),F2),A3)) = aa(set(B),set(A),image2(B,A,F2),A3) ).

% Setcompr_eq_image
tff(fact_5485_image__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: 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)),A3),B3)) = 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),A3)),aa(set(B),set(A),image2(B,A,F2),B3)) ).

% image_Un
tff(fact_5486_closed__diagonal,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_yj(product_prod(A,A),bool))) ) ).

% closed_diagonal
tff(fact_5487_finite_Omono,axiom,
    ! [A: $tType] : pp(aa(fun(fun(set(A),bool),fun(set(A),bool)),bool,order_mono(fun(set(A),bool),fun(set(A),bool)),aTP_Lamp_yk(fun(set(A),bool),fun(set(A),bool)))) ).

% finite.mono
tff(fact_5488_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)))) ) ).

% distrib_sup_le
tff(fact_5489_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)))) ) ).

% distrib_inf_le
tff(fact_5490_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F2: fun(A,B),A3: A,B3: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A3)),aa(A,B,F2,B3))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)))) ) ) ).

% mono_sup
tff(fact_5491_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5492_singleton__Un__iff,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_5493_Un__singleton__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),X: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_5494_insert__is__Un,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))),A3) ).

% insert_is_Un
tff(fact_5495_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_5496_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),A3)) ) ).

% Un_Int_assoc_eq
tff(fact_5497_Diff__subset__conv,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),C3))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3))) ) ).

% Diff_subset_conv
tff(fact_5498_Diff__partition,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_5499_Un__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = A3 ).

% Un_Diff_Int
tff(fact_5500_Int__Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = A3 ).

% Int_Diff_Un
tff(fact_5501_Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C3)) ).

% Diff_Int
tff(fact_5502_Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C3)) ).

% Diff_Un
tff(fact_5503_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A2: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q))))
     => ( ! [X4: A,Y2: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),B2)),transitive_rtrancl(A,P)))
           => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X4)),Q))
             => ( Y2 = X4 ) ) )
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,P))) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_5504_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A2: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q))))
     => ( ! [X4: A,Y2: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),X4)),transitive_rtrancl(A,P)))
           => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y2)),Q))
             => ( X4 = Y2 ) ) )
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,P))) ) ) ).

% rtrancl_Un_separatorE
tff(fact_5505_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_5506_Compl__partition,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = top_top(set(A)) ).

% Compl_partition
tff(fact_5507_Compl__partition2,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),A3) = top_top(set(A)) ).

% Compl_partition2
tff(fact_5508_eventually__finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( eventually(set(A),P,finite5375528669736107172at_top(A,A3))
      <=> pp(aa(set(A),bool,P,A3)) ) ) ).

% eventually_finite_subsets_at_top_finite
tff(fact_5509_Compl__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Compl_Un
tff(fact_5510_Compl__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Compl_Int
tff(fact_5511_closed__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_yl(product_prod(A,A),bool))) ) ).

% closed_superdiagonal
tff(fact_5512_closed__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_ym(product_prod(A,A),bool))) ) ).

% closed_subdiagonal
tff(fact_5513_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_yn(product_prod(A,A),bool))) ) ).

% open_subdiagonal
tff(fact_5514_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_yo(product_prod(A,A),bool))) ) ).

% open_superdiagonal
tff(fact_5515_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => ( aa(list(A),set(A),set2(A),Zs) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ) ) ).

% set_shuffles
tff(fact_5516_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_yp(fun(B,A),fun(A,bool),F2)) = aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_5517_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,bool)),Net: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( eventually(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_yq(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A3),P),Net)
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_xw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X3),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_5518_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,bool)),Net: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_xw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X4),Net) )
       => eventually(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_yq(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A3),P),Net) ) ) ).

% eventually_ball_finite
tff(fact_5519_mono__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A3: set(A),B3: set(A)] :
      ( pp(aa(fun(set(A),set(B)),bool,order_mono(set(A),set(B)),F2))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(A),set(B),F2,A3)),aa(set(A),set(B),F2,B3))),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))) ) ).

% mono_Un
tff(fact_5520_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% sup_shunt
tff(fact_5521_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z))) ) ) ).

% shunt1
tff(fact_5522_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z))) ) ) ).

% shunt2
tff(fact_5523_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P2: A,Q4: A,R2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q4),R2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P2),aa(A,A,uminus_uminus(A),Q4))),R2)) ) ) ).

% sup_neg_inf
tff(fact_5524_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A2: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),X) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),X) = top_top(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),Y) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Y) = top_top(A) )
               => ( X = Y ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_5525_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X4: A,Y2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),Y2)),A3)) ) )
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Sup_Sup(A),A3)),A3)) ) ) ) ) ).

% finite_Sup_in
tff(fact_5526_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))) ) ).

% less_eq_Inf_inter
tff(fact_5527_option_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,Option: option(A),F1: B,F22: fun(A,B)] :
      ( ( ( Option = none(A) )
       => ( case_option(B,A,F1,F22,Option) = F1 ) )
      & ( ( Option != none(A) )
       => ( case_option(B,A,F1,F22,Option) = aa(A,B,F22,aa(option(A),A,the2(A),Option)) ) ) ) ).

% option.case_eq_if
tff(fact_5528_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_5529_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_5530_card__Un__le,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)))) ).

% card_Un_le
tff(fact_5531_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(B,set(A))] : 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)),aa(set(B),set(set(A)),image2(B,set(A),F2),bot_bot(set(B))))) = A3 ).

% Union_image_empty
tff(fact_5532_case__optionE,axiom,
    ! [A: $tType,P: bool,Q: fun(A,bool),X: option(A)] :
      ( pp(case_option(bool,A,P,Q,X))
     => ( ( ( X = none(A) )
         => ~ pp(P) )
       => ~ ! [Y2: A] :
              ( ( X = aa(A,option(A),some(A),Y2) )
             => ~ pp(aa(A,bool,Q,Y2)) ) ) ) ).

% case_optionE
tff(fact_5533_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_5534_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_5535_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),set_ord_atLeast(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_5536_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_atMost(A,L)),set_or3652927894154168847AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_5537_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_yr(set(A),fun(A,bool),A3))) ) ).

% Sup_eq_Inf
tff(fact_5538_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ys(set(A),fun(A,bool),A3))) ) ).

% Inf_eq_Sup
tff(fact_5539_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),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_5540_Pow__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : pow(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow(A,A3)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2)),pow(A,A3))) ).

% Pow_insert
tff(fact_5541_UN__extend__simps_I2_J,axiom,
    ! [D: $tType,C: $tType,C3: set(C),A3: fun(C,set(D)),B3: set(D)] :
      ( ( ( C3 = bot_bot(set(C)) )
       => ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A3),C3))),B3) = B3 ) )
      & ( ( C3 != bot_bot(set(C)) )
       => ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A3),C3))),B3) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_xz(fun(C,set(D)),fun(set(D),fun(C,set(D))),A3),B3)),C3)) ) ) ) ).

% UN_extend_simps(2)
tff(fact_5542_UN__extend__simps_I3_J,axiom,
    ! [E: $tType,F: $tType,C3: set(F),A3: set(E),B3: fun(F,set(E))] :
      ( ( ( C3 = bot_bot(set(F)) )
       => ( aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A3),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),B3),C3))) = A3 ) )
      & ( ( C3 != bot_bot(set(F)) )
       => ( aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A3),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),B3),C3))) = aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F),set(set(E)),image2(F,set(E),aa(fun(F,set(E)),fun(F,set(E)),aTP_Lamp_ya(set(E),fun(fun(F,set(E)),fun(F,set(E))),A3),B3)),C3)) ) ) ) ).

% UN_extend_simps(3)
tff(fact_5543_Inter__Un__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B3)))) ).

% Inter_Un_subset
tff(fact_5544_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),X) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_5545_eventually__finite__subsets__at__top,axiom,
    ! [A: $tType,P: fun(set(A),bool),A3: set(A)] :
      ( eventually(set(A),P,finite5375528669736107172at_top(A,A3))
    <=> ? [X8: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X8))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X8),A3))
          & ! [Y7: set(A)] :
              ( ( pp(aa(set(A),bool,finite_finite2(A),Y7))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X8),Y7))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Y7),A3)) )
             => pp(aa(set(A),bool,P,Y7)) ) ) ) ).

% eventually_finite_subsets_at_top
tff(fact_5546_cSup__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X6)
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X6)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(set(A),A,complete_Sup_Sup(A),X6)) ) ) ) ) ).

% cSup_insert
tff(fact_5547_cSup__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( condit941137186595557371_above(A,X6)
         => ( ( ( X6 = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X6)) = A2 ) )
            & ( ( X6 != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X6)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(set(A),A,complete_Sup_Sup(A),X6)) ) ) ) ) ) ).

% cSup_insert_If
tff(fact_5548_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_5549_sum_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B3)) ) ) ) ) ).

% sum.union_inter
tff(fact_5550_prod_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ).

% prod.union_inter
tff(fact_5551_card__Un__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,finite_finite2(A),B3))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) = 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)),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% card_Un_Int
tff(fact_5552_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_5553_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_5554_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_5555_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)),set_ord_lessThan(A,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_ord_atMost(A,U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_5556_cInf__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A3)
           => ( ( B3 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,B3)
               => ( 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)),A3),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_5557_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_5558_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => ( ( B3 != 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)),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B3)) ) ) ) ) ) ) ).

% Max.union
tff(fact_5559_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_atMost(A,L)),set_or5935395276787703475ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_5560_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_5561_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_5562_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_yt(list(A),fun(A,bool),Xs)) ).

% set_conv_nth
tff(fact_5563_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(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_5564_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_ll(A,fun(nat,A),B3)),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ).

% SUP_nat_binary
tff(fact_5565_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),set_ord_atLeast(A,U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_5566_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G),A3))
             => ( 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),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),A3))) = 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_yu(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_5567_option_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,case_option(B,A,F1,F22,Option)))
    <=> ~ ( ( ( Option = none(A) )
            & ~ pp(aa(B,bool,P,F1)) )
          | ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
            & ~ pp(aa(B,bool,P,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel_asm
tff(fact_5568_option_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,case_option(B,A,F1,F22,Option)))
    <=> ( ( ( Option = none(A) )
         => pp(aa(B,bool,P,F1)) )
        & ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
         => pp(aa(B,bool,P,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel
tff(fact_5569_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),G: fun(A,B),B3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( inj_on(A,B,G,B3)
       => ( ( 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),A3)),aa(set(A),set(B),image2(A,B,G),B3)) = 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_yv(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_5570_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_yw(A,fun(A,bool)),aTP_Lamp_yx(A,fun(A,bool))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_5571_finite__subsets__at__top__def,axiom,
    ! [A: $tType,A3: set(A)] : finite5375528669736107172at_top(A,A3) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image2(set(A),filter(set(A)),aTP_Lamp_yz(set(A),fun(set(A),filter(set(A))),A3)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_za(set(A),fun(set(A),bool),A3)))) ).

% finite_subsets_at_top_def
tff(fact_5572_rtrancl__insert,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R2)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_zb(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A2),B2),R2)))) ).

% rtrancl_insert
tff(fact_5573_trancl__insert2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_zc(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A2),B2),R2)))) ).

% trancl_insert2
tff(fact_5574_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( condit941137186595557371_above(A,A3)
         => ( condit941137186595557371_above(A,B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) != bot_bot(set(A)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_5575_cSUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),A2: B] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( 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),A2),A3))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,A2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))) ) ) ) ) ).

% cSUP_insert
tff(fact_5576_sum_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)))
                 => ( aa(B,A,G,X4) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B3)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_5577_sum__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A3: set(B),B3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) ) ) ) ) ).

% sum_Un
tff(fact_5578_sum_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3) = bot_bot(set(B)) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B3)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_5579_prod_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)))
                 => ( aa(B,A,G,X4) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_5580_prod_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3) = bot_bot(set(B)) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_5581_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),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_5582_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% sum_Un2
tff(fact_5583_sum_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B3),A3)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) ) ) ) ) ).

% sum.union_diff2
tff(fact_5584_prod_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B3),A3)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) ) ) ) ) ).

% prod.union_diff2
tff(fact_5585_card__Un__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,finite_finite2(A),B3))
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = 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)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_5586_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
    <=> ( inj_on(A,B,F2,A3)
        & inj_on(A,B,F2,B3)
        & ( 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),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_5587_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),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_5588_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_5589_cINF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F2: fun(B,A),B3: set(B)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
           => ( ( B3 != bot_bot(set(B)) )
             => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),B3))
               => ( aa(set(A),A,complete_Inf_Inf(A),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)),A3),B3))) = 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,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),B3))) ) ) ) ) ) ) ).

% cINF_union
tff(fact_5590_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),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_5591_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A)] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S2)
           => ( aa(set(A),A,complete_Inf_Inf(A),S2) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_zd(set(A),fun(A,bool),S2))) ) ) ) ) ).

% cInf_cSup
tff(fact_5592_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A)] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S2)
           => ( aa(set(A),A,complete_Sup_Sup(A),S2) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ze(set(A),fun(A,bool),S2))) ) ) ) ) ).

% cSup_cInf
tff(fact_5593_sum__Un__nat,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,finite_finite2(A),B3))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B3))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_5594_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_5595_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),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_5596_trancl__insert,axiom,
    ! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] : transitive_trancl(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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_zb(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Y),X),R2)))) ).

% trancl_insert
tff(fact_5597_prod__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [A3: set(B),B3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( pp(aa(set(B),bool,finite_finite2(B),B3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)))
                 => ( aa(B,A,F2,X4) != zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),B3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) ) ) ) ) ) ).

% prod_Un
tff(fact_5598_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 )
        & ! [X3: vEBT_VEBT] :
            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
           => vEBT_VEBT_valid(X3,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
        & vEBT_VEBT_valid(Summary,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
        & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_xu(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg),TreeList),Summary)),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5599_funpow__inj__finite,axiom,
    ! [A: $tType,P2: fun(A,A),X: A] :
      ( inj_on(A,A,P2,top_top(set(A)))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_zf(fun(A,A),fun(A,fun(A,bool)),P2),X))))
       => ~ ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),P2),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_5600_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,set(B)),A3: set(B),F4: filter(A)] :
      ( filterlim(A,set(B),F2,finite5375528669736107172at_top(B,A3),F4)
    <=> ! [X8: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),X8))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X8),A3)) )
         => eventually(A,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_zg(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),F2),A3),X8),F4) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_5601_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_valid(X,Xa2)
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa2 != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_VEBT_valid(X3,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                    & vEBT_VEBT_valid(Summary3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_xu(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary3)),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5602_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_valid(X,Xa2)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X2: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                     => vEBT_VEBT_valid(X2,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                  & vEBT_VEBT_valid(Summary3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                  & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                  & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_xu(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary3)),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_5603_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2))
               => ( Xa2 = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary3)),Xa2))
                 => ( ( Deg2 = Xa2 )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => vEBT_VEBT_valid(X4,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                    & vEBT_VEBT_valid(Summary3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_xu(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary3)),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_5604_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_valid(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2))
               => ( Xa2 != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary3)),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_VEBT_valid(X2,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                      & vEBT_VEBT_valid(Summary3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_xu(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary3)),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_5605_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_valid(X,Xa2)
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa2 = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                         => vEBT_VEBT_valid(X3,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                      & vEBT_VEBT_valid(Summary3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_xu(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList2),Summary3)),Mima)) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary3)),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_5606_sup__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ).

% sup_set_def
tff(fact_5607_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_5608_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B)),X2: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),S2)),X2),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R),S2))) ) ).

% sup_Un_eq2
tff(fact_5609_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)] :
          ( pp(aa(fun(A,fun(B,C)),bool,order_mono(A,fun(B,C)),Q))
         => pp(aa(fun(A,fun(D,C)),bool,order_mono(A,fun(D,C)),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_zh(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F2))) ) ) ).

% mono_compose
tff(fact_5610_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_zi(set(set(A)),fun(set(A),bool),A3))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A3)))) ) ).

% Sup_Inf_le
tff(fact_5611_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A3))),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),bool),set(set(A)),collect(set(A)),aTP_Lamp_zj(set(set(A)),fun(set(A),bool),A3)))))) ) ).

% Inf_Sup_le
tff(fact_5612_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A3: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A3))),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),bool),set(set(A)),collect(set(A)),aTP_Lamp_zk(set(set(A)),fun(set(A),bool),A3)))))) ) ).

% finite_Inf_Sup
tff(fact_5613_Pow__Compl,axiom,
    ! [A: $tType,A3: set(A)] : pow(A,aa(set(A),set(A),uminus_uminus(set(A)),A3)) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_zl(set(A),fun(set(A),bool),A3)) ).

% Pow_Compl
tff(fact_5614_Union__maximal__sets,axiom,
    ! [A: $tType,F13: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),F13))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_zm(set(set(A)),fun(set(A),bool),F13))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F13) ) ) ).

% Union_maximal_sets
tff(fact_5615_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ! [X4: product_prod(list(A),list(A))] :
          ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),X4),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys))))
         => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_zn(list(A),fun(list(A),bool))),X4)) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ( aa(list(list(A)),list(A),concat(A),Xs) = aa(list(list(A)),list(A),concat(A),Ys) )
        <=> ( Xs = Ys ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_5616_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ( aa(list(list(A)),list(A),concat(A),Xs) = aa(list(list(A)),list(A),concat(A),Ys) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ! [X4: product_prod(list(A),list(A))] :
              ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),X4),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys))))
             => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_zn(list(A),fun(list(A),bool))),X4)) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_5617_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X3: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),X3),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys))))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),fequal(A)),X3)) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_5618_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C3: set(A),B3: set(A),X: A] :
      ( inj_on(A,B,G,C3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))
       => pp(aa(set(fun(B,A)),bool,aa(fun(B,A),fun(set(fun(B,A)),bool),member(fun(B,A)),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_zo(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C3),X)),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_5619_min__weak__def,axiom,
    fun_min_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),min_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).

% min_weak_def
tff(fact_5620_Pow__fold,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pow(A,A3) = finite_fold(A,set(set(A)),aTP_Lamp_zp(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)))),A3) ) ) ).

% Pow_fold
tff(fact_5621_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_5622_fold__infinite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( finite_fold(A,B,F2,Z,A3) = Z ) ) ).

% fold_infinite
tff(fact_5623_fold__closed__eq,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),F2: fun(A,fun(B,B)),G: fun(A,fun(B,B)),Z: B] :
      ( ! [A4: A,B4: B] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),B3))
           => ( aa(B,B,aa(A,fun(B,B),F2,A4),B4) = aa(B,B,aa(A,fun(B,B),G,A4),B4) ) ) )
     => ( ! [A4: A,B4: B] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),B3))
             => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(B,B,aa(A,fun(B,B),G,A4),B4)),B3)) ) )
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Z),B3))
         => ( finite_fold(A,B,F2,Z,A3) = finite_fold(A,B,G,Z,A3) ) ) ) ) ).

% fold_closed_eq
tff(fact_5624_folding__on_Oeq__fold,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Z: B,A3: set(A)] :
      ( finite_folding_on(A,B,S2,F2)
     => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A3) = finite_fold(A,B,F2,Z,A3) ) ) ).

% folding_on.eq_fold
tff(fact_5625_union__fold__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = finite_fold(A,set(A),insert2(A),B3,A3) ) ) ).

% union_fold_insert
tff(fact_5626_sup__Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B3) = finite_fold(A,A,sup_sup(A),B3,A3) ) ) ) ).

% sup_Sup_fold_sup
tff(fact_5627_inf__Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A3)),B3) = finite_fold(A,A,inf_inf(A),B3,A3) ) ) ) ).

% inf_Inf_fold_inf
tff(fact_5628_fold__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A3: set(A),F2: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G,A3)
     => ( finite_fold(B,C,F2,Z,aa(set(A),set(B),image2(A,B,G),A3)) = finite_fold(A,C,aa(fun(A,B),fun(A,fun(C,C)),comp(B,fun(C,C),A,F2),G),Z,A3) ) ) ).

% fold_image
tff(fact_5629_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,complete_Sup_Sup(A),A3) = finite_fold(A,A,sup_sup(A),bot_bot(A),A3) ) ) ) ).

% Sup_fold_sup
tff(fact_5630_Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,complete_Inf_Inf(A),A3) = finite_fold(A,A,inf_inf(A),top_top(A),A3) ) ) ) ).

% Inf_fold_inf
tff(fact_5631_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = 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),A3) ) ).

% sum.eq_fold
tff(fact_5632_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,A,ord_max(A),X,A3) ) ) ) ).

% Max.eq_fold
tff(fact_5633_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( aa(set(A),set(B),image2(A,B,F2),A3) = finite_fold(A,set(B),aTP_Lamp_zq(fun(A,B),fun(A,fun(set(B),set(B))),F2),bot_bot(set(B)),A3) ) ) ).

% image_fold_insert
tff(fact_5634_sup__SUP__fold__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: A,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F2),B3,A3) ) ) ) ).

% sup_SUP_fold_sup
tff(fact_5635_inf__INF__fold__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: A,F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F2),B3,A3) ) ) ) ).

% inf_INF_fold_inf
tff(fact_5636_SUP__fold__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F2),bot_bot(A),A3) ) ) ) ).

% SUP_fold_sup
tff(fact_5637_INF__fold__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F2),top_top(A),A3) ) ) ) ).

% INF_fold_inf
tff(fact_5638_max__weak__def,axiom,
    fun_max_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),max_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).

% max_weak_def
tff(fact_5639_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A15: set(B),B1: set(A),F22: fun(C,D),B22: set(C),A25: set(D)] :
      ( ( aa(set(B),set(A),image2(B,A,F1),A15) = B1 )
     => ( inj_on(C,D,F22,B22)
       => ( pp(aa(set(D),bool,aa(set(D),fun(set(D),bool),ord_less_eq(set(D)),aa(set(C),set(D),image2(C,D,F22),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,F1,F22)),bNF_Wellorder_Func(D,B,A25,A15)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_5640_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R2)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X3: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),X3)) ) ) ) ).

% listrel_iff_zip
tff(fact_5641_listrel__rtrancl__refl,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),listrel(A,A,transitive_rtrancl(A,R2)))) ).

% listrel_rtrancl_refl
tff(fact_5642_Ball__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(A,bool,P,X3)) )
      <=> pp(finite_fold(A,bool,aTP_Lamp_zr(fun(A,bool),fun(A,fun(bool,bool)),P),fTrue,A3)) ) ) ).

% Ball_fold
tff(fact_5643_listrel_ONil,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B))),listrel(A,B,R2))) ).

% listrel.Nil
tff(fact_5644_listrel__Nil1,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),R2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xs)),listrel(A,B,R2)))
     => ( Xs = nil(B) ) ) ).

% listrel_Nil1
tff(fact_5645_listrel__Nil2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),R2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),nil(B))),listrel(A,B,R2)))
     => ( Xs = nil(A) ) ) ).

% listrel_Nil2
tff(fact_5646_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R2)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel_eq_len
tff(fact_5647_listrel__rtrancl__trans,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Zs: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel(A,A,transitive_rtrancl(A,R2))))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),listrel(A,A,transitive_rtrancl(A,R2))))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs)),listrel(A,A,transitive_rtrancl(A,R2)))) ) ) ).

% listrel_rtrancl_trans
tff(fact_5648_listrel__mono,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R2),S))
     => pp(aa(set(product_prod(list(A),list(B))),bool,aa(set(product_prod(list(A),list(B))),fun(set(product_prod(list(A),list(B))),bool),ord_less_eq(set(product_prod(list(A),list(B)))),listrel(A,B,R2)),listrel(A,B,S))) ) ).

% listrel_mono
tff(fact_5649_max__ext__additive,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),R: set(product_prod(A,A)),C3: set(A),D6: set(A)] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A3),B3)),max_ext(A,R)))
     => ( pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),C3),D6)),max_ext(A,R)))
       => pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),D6))),max_ext(A,R))) ) ) ).

% max_ext_additive
tff(fact_5650_card_Oeq__fold,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = finite_fold(A,nat,aTP_Lamp_mg(A,fun(nat,nat)),zero_zero(nat),A3) ).

% card.eq_fold
tff(fact_5651_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),nil(A),A3) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_5652_listrel__reflcl__if__listrel1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel(A,A,transitive_rtrancl(A,R2)))) ) ).

% listrel_reflcl_if_listrel1
tff(fact_5653_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : listrel(A,A,transitive_rtrancl(A,R2)) = transitive_rtrancl(list(A),listrel1(A,R2)) ).

% listrel_rtrancl_eq_rtrancl_listrel1
tff(fact_5654_rtrancl__listrel1__if__listrel,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel(A,A,R2)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R2)))) ) ).

% rtrancl_listrel1_if_listrel
tff(fact_5655_Func__map,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,G: fun(A,B),A25: set(A),A15: set(B),F1: fun(B,C),B1: set(C),F22: fun(D,A),B22: set(D)] :
      ( pp(aa(set(fun(A,B)),bool,aa(fun(A,B),fun(set(fun(A,B)),bool),member(fun(A,B)),G),bNF_Wellorder_Func(A,B,A25,A15)))
     => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F1),A15)),B1))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),image2(D,A,F22),B22)),A25))
         => pp(aa(set(fun(D,C)),bool,aa(fun(D,C),fun(set(fun(D,C)),bool),member(fun(D,C)),aa(fun(A,B),fun(D,C),bNF_We4925052301507509544nc_map(D,B,C,A,B22,F1,F22),G)),bNF_Wellorder_Func(D,C,B22,B1))) ) ) ) ).

% Func_map
tff(fact_5656_listrel__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),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_5657_Id__on__fold,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( id_on(A,A3) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_zs(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A3) ) ) ).

% Id_on_fold
tff(fact_5658_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R2)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N3)),aa(nat,B,nth(B,Ys),N3))),R2)) ) ) ) ).

% listrel_iff_nth
tff(fact_5659_max__ext_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y5: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),Y5))
       => ( ( Y5 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => ? [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),Y5))
                    & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),R)) ) )
           => pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X6),Y5)),max_ext(A,R))) ) ) ) ) ).

% max_ext.max_extI
tff(fact_5660_max__ext_Osimps,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A1))
        & pp(aa(set(A),bool,finite_finite2(A),A22))
        & ( A22 != bot_bot(set(A)) )
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A1))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A22))
                & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),R)) ) ) ) ) ).

% max_ext.simps
tff(fact_5661_max__ext_Ocases,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R)))
     => ~ ( pp(aa(set(A),bool,finite_finite2(A),A1))
         => ( pp(aa(set(A),bool,finite_finite2(A),A22))
           => ( ( A22 != bot_bot(set(A)) )
             => ~ ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A1))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A22))
                        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa4)),R)) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_5662_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_zt(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Max.eq_fold'
tff(fact_5663_Func__is__emp,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( ( bNF_Wellorder_Func(A,B,A3,B3) = bot_bot(set(fun(A,B))) )
    <=> ( ( A3 != bot_bot(set(A)) )
        & ( B3 = bot_bot(set(B)) ) ) ) ).

% Func_is_emp
tff(fact_5664_Func__non__emp,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A3: set(B)] :
      ( ( B3 != bot_bot(set(A)) )
     => ( bNF_Wellorder_Func(B,A,A3,B3) != bot_bot(set(fun(B,A))) ) ) ).

% Func_non_emp
tff(fact_5665_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B3: set(A),X: B,A3: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( 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_zu(B,fun(A,set(product_prod(B,A))),X)),B3))),A3) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_zv(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X),A3,B3) ) ) ).

% fold_union_pair
tff(fact_5666_Set__filter__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( filter3(A,P,A3) = finite_fold(A,set(A),aTP_Lamp_zw(fun(A,bool),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A3) ) ) ).

% Set_filter_fold
tff(fact_5667_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A)] : pp(aa(fun(fun(A,bool),fun(A,bool)),bool,order_mono(fun(A,bool),fun(A,bool)),aTP_Lamp_zx(fun(A,A),fun(fun(A,bool),fun(A,bool)),F2))) ) ).

% iteratesp.mono
tff(fact_5668_max__ext__def,axiom,
    ! [A: $tType,X2: set(product_prod(A,A))] : max_ext(A,X2) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),max_extp(A,aTP_Lamp_wl(set(product_prod(A,A)),fun(A,fun(A,bool)),X2)))) ).

% max_ext_def
tff(fact_5669_member__filter,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),filter3(A,P,A3)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
        & pp(aa(A,bool,P,X)) ) ) ).

% member_filter
tff(fact_5670_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,bool),A3: set(A)] : filter3(A,P,A3) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_zy(fun(A,bool),fun(set(A),fun(A,bool)),P),A3)) ).

% Set.filter_def
tff(fact_5671_chain__subset,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool)),A3: set(A),B3: set(A)] :
      ( comple1602240252501008431_chain(A,Ord,A3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => comple1602240252501008431_chain(A,Ord,B3) ) ) ).

% chain_subset
tff(fact_5672_chain__empty,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).

% chain_empty
tff(fact_5673_finite__filter,axiom,
    ! [A: $tType,S2: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => pp(aa(set(A),bool,finite_finite2(A),filter3(A,P,S2))) ) ).

% finite_filter
tff(fact_5674_ccpo__Sup__upper,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: set(A),X: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% ccpo_Sup_upper
tff(fact_5675_ccpo__Sup__least,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: set(A),Z: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Z)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z)) ) ) ) ).

% ccpo_Sup_least
tff(fact_5676_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_5677_inter__Set__filter,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = filter3(A,aTP_Lamp_a(set(A),fun(A,bool),A3),B3) ) ) ).

% inter_Set_filter
tff(fact_5678_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X2: set(A),Xa: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,aTP_Lamp_wl(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X2),Xa))
    <=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X2),Xa)),max_ext(A,R))) ) ).

% max_extp_max_ext_eq
tff(fact_5679_max__extp__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X: set(A),Y: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R2),X),Y))
    <=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X),Y)),max_ext(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R2))))) ) ).

% max_extp_eq
tff(fact_5680_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
         => ( pp(aa(set(A),bool,finite_finite2(A),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Sup_Sup(A),A3)),A3)) ) ) ) ) ).

% in_chain_finite
tff(fact_5681_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_zz(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Sup_fin.eq_fold'
tff(fact_5682_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aaa(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Inf_fin.eq_fold'
tff(fact_5683_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = A2 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_5684_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% Sup_fin.singleton
tff(fact_5685_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% Inf_fin.singleton
tff(fact_5686_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A2) = A2 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_5687_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_5688_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_5689_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),A3))) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_5690_Sup__fin__Max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).

% Sup_fin_Max
tff(fact_5691_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A2)) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_5692_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A3))) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_5693_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_5694_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A3) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_5695_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)))
             => ! [A10: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A10),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A10)) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_5696_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A4)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3))) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_5697_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),X))
             => ! [A10: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A10),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A10),X)) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_5698_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),X)) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_5699_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X3)) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_5700_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),X))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),X)) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_5701_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_5702_cSup__eq__Sup__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = aa(set(A),A,lattic5882676163264333800up_fin(A),X6) ) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_5703_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(set(A),A,complete_Inf_Inf(A),A3) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_5704_cInf__eq__Inf__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = aa(set(A),A,lattic7752659483105999362nf_fin(A),X6) ) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_5705_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_5706_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_5707_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3))) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_5708_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3))) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_5709_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H: fun(A,A),N4: set(A)] :
          ( ! [X4: A,Y2: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),Y2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X4)),aa(A,A,H,Y2))
         => ( pp(aa(set(A),bool,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_5710_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H: fun(A,A),N4: set(A)] :
          ( ! [X4: A,Y2: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),Y2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X4)),aa(A,A,H,Y2))
         => ( pp(aa(set(A),bool,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_5711_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( B3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_5712_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( B3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A3) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_5713_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X4: A,Y2: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A3)) ) ) ) ) ).

% Inf_fin.closed
tff(fact_5714_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( A3 != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_5715_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( A3 != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_5716_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X4: A,Y2: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),A3)) ) ) ) ) ).

% Sup_fin.closed
tff(fact_5717_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => ( ( B3 != 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)),A3),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_5718_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => ( ( B3 != 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)),A3),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_5719_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,A,inf_inf(A),X,A3) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_5720_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,A,sup_sup(A),X,A3) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_5721_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aab(set(A),fun(A,fun(A,bool)),A3),X))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_5722_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aac(set(A),fun(set(A),fun(A,bool)),A3),B3))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_5723_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aad(set(A),fun(A,fun(A,bool)),A3),X))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_5724_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aae(set(A),fun(set(A),fun(A,bool)),A3),B3))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_5725_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Inf_fin.remove
tff(fact_5726_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = 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),X),A3)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_5727_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Sup_fin.remove
tff(fact_5728_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = 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),X),A3)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_5729_flat__lub__def,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A3) = B2 ) )
      & ( ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A3) = the(A,aa(A,fun(A,bool),aTP_Lamp_aaf(set(A),fun(A,fun(A,bool)),A3),B2)) ) ) ) ).

% flat_lub_def
tff(fact_5730_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B)),X: product_prod(C,A),X6: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S2))
     => ( 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)),X),bot_bot(set(product_prod(C,A)))),S2)),X6) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_aag(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),X6,S2) ) ) ).

% insert_relcomp_union_fold
tff(fact_5731_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B3))
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_aah(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B3)) ) ).

% comp_fun_commute_product_fold
tff(fact_5732_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_5733_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_5734_finite__relcomp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R))
     => ( pp(aa(set(product_prod(B,C)),bool,finite_finite2(product_prod(B,C)),S2))
       => pp(aa(set(product_prod(A,C)),bool,finite_finite2(product_prod(A,C)),relcomp(A,B,C,R,S2))) ) ) ).

% finite_relcomp
tff(fact_5735_comp__fun__commute__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F2)
    <=> ! [Y4: A,X3: 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,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,Y4)) ) ).

% comp_fun_commute_def
tff(fact_5736_comp__fun__commute_Ocomp__fun__commute,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),Y: A,X: A] :
      ( finite6289374366891150609ommute(A,B,F2)
     => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),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,Y)) ) ) ).

% comp_fun_commute.comp_fun_commute
tff(fact_5737_comp__fun__commute_Ointro,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( ! [Y2: A,X4: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y2)),aa(A,fun(B,B),F2,X4)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X4)),aa(A,fun(B,B),F2,Y2))
     => finite6289374366891150609ommute(A,B,F2) ) ).

% comp_fun_commute.intro
tff(fact_5738_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_5739_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_aai(fun(B,B),fun(A,fun(B,B)),F2)) ).

% comp_fun_commute_const
tff(fact_5740_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A2: A,C2: B,R2: set(product_prod(A,C)),S: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),C2)),relcomp(A,C,B,R2,S)))
     => ~ ! [B4: C] :
            ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A2),B4)),R2))
           => ~ pp(aa(set(product_prod(C,B)),bool,aa(product_prod(C,B),fun(set(product_prod(C,B)),bool),member(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),C2)),S)) ) ) ).

% relcompEpair
tff(fact_5741_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R2: set(product_prod(A,C)),S: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),Xz),relcomp(A,C,B,R2,S)))
     => ~ ! [X4: A,Y2: C,Z3: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z3) )
           => ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X4),Y2)),R2))
             => ~ pp(aa(set(product_prod(C,B)),bool,aa(product_prod(C,B),fun(set(product_prod(C,B)),bool),member(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y2),Z3)),S)) ) ) ) ).

% relcompE
tff(fact_5742_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R2: set(product_prod(A,B)),C2: C,S: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),R2))
     => ( pp(aa(set(product_prod(B,C)),bool,aa(product_prod(B,C),fun(set(product_prod(B,C)),bool),member(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C2)),S))
       => pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A2),C2)),relcomp(A,B,C,R2,S))) ) ) ).

% relcomp.relcompI
tff(fact_5743_relcomp_Osimps,axiom,
    ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R2: set(product_prod(A,B)),S: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22)),relcomp(A,B,C,R2,S)))
    <=> ? [A7: A,B7: B,C5: C] :
          ( ( A1 = A7 )
          & ( A22 = C5 )
          & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7)),R2))
          & pp(aa(set(product_prod(B,C)),bool,aa(product_prod(B,C),fun(set(product_prod(B,C)),bool),member(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B7),C5)),S)) ) ) ).

% relcomp.simps
tff(fact_5744_relcomp_Ocases,axiom,
    ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R2: set(product_prod(A,B)),S: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22)),relcomp(A,B,C,R2,S)))
     => ~ ! [B4: B] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B4)),R2))
           => ~ pp(aa(set(product_prod(B,C)),bool,aa(product_prod(B,C),fun(set(product_prod(B,C)),bool),member(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),A22)),S)) ) ) ).

% relcomp.cases
tff(fact_5745_union__comp__emptyL,axiom,
    ! [A: $tType,A3: set(product_prod(A,A)),C3: set(product_prod(A,A)),B3: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A3,C3) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,B3,C3) = bot_bot(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))),sup_sup(set(product_prod(A,A))),A3),B3),C3) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyL
tff(fact_5746_union__comp__emptyR,axiom,
    ! [A: $tType,A3: set(product_prod(A,A)),B3: set(product_prod(A,A)),C3: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A3,B3) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,A3,C3) = bot_bot(set(product_prod(A,A))) )
       => ( relcomp(A,A,A,A3,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),B3),C3)) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyR
tff(fact_5747_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P: fun(A,bool)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_zw(fun(A,bool),fun(A,fun(set(A),set(A))),P)) ).

% comp_fun_commute_filter_fold
tff(fact_5748_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_aaj(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_5749_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S: set(product_prod(C,B))] : relcomp(A,C,B,R2,S) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_aak(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),R2),S))) ).

% relcomp_unfold
tff(fact_5750_max__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S2)),R))
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),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,S2)),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)),aa(set(A),fun(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_5751_min__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S2)),R))
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),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,S2)),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)),aa(set(A),fun(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_5752_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R))
     => ( pp(aa(set(product_prod(B,C)),bool,finite_finite2(product_prod(B,C)),S2))
       => ( relcomp(A,B,C,R,S2) = 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_aam(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S2)),bot_bot(set(product_prod(A,C))),R) ) ) ) ).

% relcomp_fold
tff(fact_5753_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S2))
     => 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_aao(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S2))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_5754_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B)),X: product_prod(C,A),R: set(product_prod(C,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S2))
     => ( 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)),X),R),S2) = 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_aag(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),relcomp(C,A,B,R,S2),S2) ) ) ).

% insert_relcomp_fold
tff(fact_5755_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: set(C),F2: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A3,F2,G) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_aap(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),A3),F2),G)) ).

% image2_def
tff(fact_5756_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_numeral
tff(fact_5757_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I: B,B3: 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),A3,I,B3)),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),A3),aa(set(B),set(B),aa(set(B),fun(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))))))),if(set(A),aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),J4),B3,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_5758_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_5759_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa(nat,nat,suc,N),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(2)
tff(fact_5760_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_5761_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_aaq(num,option(num)),bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_5762_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_5763_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C2: B,P: fun(fun(A,B),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_aar(fun(A,B),fun(B,fun(A,bool)),F2),C2))))
     => ( pp(aa(fun(A,B),bool,P,aTP_Lamp_ku(B,fun(A,B),C2)))
       => ( ! [A4: A,B4: B,F3: fun(A,B)] :
              ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_aas(B,fun(fun(A,B),fun(A,bool)),C2),F3))))
             => ( ( aa(A,B,F3,A4) = C2 )
               => ( ( B4 != C2 )
                 => ( pp(aa(fun(A,B),bool,P,F3))
                   => pp(aa(fun(A,B),bool,P,fun_upd(A,B,F3,A4,B4))) ) ) ) )
         => pp(aa(fun(A,B),bool,P,F2)) ) ) ) ).

% finite_update_induct
tff(fact_5764_comp__fun__commute__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_kg(A,A))) ) ).

% comp_fun_commute_insort
tff(fact_5765_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num,Q4: num] :
          ( ( bit_take_bit_num(M,N) = aa(num,option(num),some(num),Q4) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q4) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_5766_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F2: fun(B,A),X: B,C2: C,G: fun(B,C),A3: set(B)] :
      ( ( B2 = aa(B,A,F2,X) )
     => ( ( C2 = aa(B,C,G,X) )
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
         => pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C2)),bNF_Greatest_image2(B,A,C,A3,F2,G))) ) ) ) ).

% image2_eqI
tff(fact_5767_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num] :
          ( ( bit_take_bit_num(M,N) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5768_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,X: B,A3: set(B),F2: fun(B,A),Y: A] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
       => ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F2,X,Y)),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) )
      & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
       => ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F2,X,Y)),A3) = aa(set(B),set(A),image2(B,A,F2),A3) ) ) ) ).

% fun_upd_image
tff(fact_5769_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_zp(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_5770_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),B3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),S2))
         => ( pp(aa(set(A),bool,finite_finite2(A),A3))
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = 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)),A3),B3)) = finite_fold(A,B,F2,finite_fold(A,B,F2,Z,A3),B3) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_5771_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_5772_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),X: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( finite_fold(A,B,F2,Z,A3) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_5773_comp__fun__commute__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),S2))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y2)),aa(A,fun(B,B),F2,X4)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X4)),aa(A,fun(B,B),F2,Y2)) ) ) )
     => finite4664212375090638736ute_on(A,B,S2,F2) ) ).

% comp_fun_commute_on.intro
tff(fact_5774_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,Y: A,G: fun(C,B)] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
         => ( aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Y)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,X)),G)) = aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,X)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Y)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_5775_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,Y: A] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),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,Y)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_5776_comp__fun__commute__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
    <=> ! [X3: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S2))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y4)),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,Y4)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_5777_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => finite4664212375090638736ute_on(A,B,S2,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_aaj(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_5778_comp__fun__commute__on_Ofun__left__comm,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,Y: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
         => ( aa(B,B,aa(A,fun(B,B),F2,Y),aa(B,B,aa(A,fun(B,B),F2,X),Z)) = aa(B,B,aa(A,fun(B,B),F2,X),aa(B,B,aa(A,fun(B,B),F2,Y),Z)) ) ) ) ) ).

% comp_fun_commute_on.fun_left_comm
tff(fact_5779_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_5780_Finite__Set_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),G: fun(A,fun(B,B)),A3: set(A),S: B,T2: B,B3: set(A)] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( finite4664212375090638736ute_on(A,B,S2,G)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
         => ( pp(aa(set(A),bool,finite_finite2(A),A3))
           => ( ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => ( aa(A,fun(B,B),F2,X4) = aa(A,fun(B,B),G,X4) ) )
             => ( ( S = T2 )
               => ( ( A3 = B3 )
                 => ( finite_fold(A,B,F2,S,A3) = finite_fold(A,B,G,T2,B3) ) ) ) ) ) ) ) ) ).

% Finite_Set.fold_cong
tff(fact_5781_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F2: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G),top_top(set(C)))),S2))
       => 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_5782_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z,A3)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_5783_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,X),Z),A3) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_5784_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z,A3)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,X),Z),A3) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_5785_image__map__upd,axiom,
    ! [B: $tType,A: $tType,X: A,A3: set(A),M: fun(A,option(B)),Y: B] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),set(option(B)),image2(A,option(B),fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y))),A3) = aa(set(A),set(option(B)),image2(A,option(B),M),A3) ) ) ).

% image_map_upd
tff(fact_5786_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),A2: B,B2: A] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),F2),top_top(set(B)))))
     => pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),fun_upd(B,option(A),F2,A2,aa(A,option(A),some(A),B2))),top_top(set(B))))) ) ).

% finite_range_updI
tff(fact_5787_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,X: A,X2: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_aat(A,option(B)),X,none(B)),X2) = none(B) ).

% empty_upd_none
tff(fact_5788_map__upd__Some__unfold,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),A2: B,B2: A,X: B,Y: A] :
      ( ( aa(B,option(A),fun_upd(B,option(A),M,A2,aa(A,option(A),some(A),B2)),X) = aa(A,option(A),some(A),Y) )
    <=> ( ( ( X = A2 )
          & ( B2 = Y ) )
        | ( ( X != A2 )
          & ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y) ) ) ) ) ).

% map_upd_Some_unfold
tff(fact_5789_map__upd__triv,axiom,
    ! [A: $tType,B: $tType,T2: fun(B,option(A)),K: B,X: A] :
      ( ( aa(B,option(A),T2,K) = aa(A,option(A),some(A),X) )
     => ( fun_upd(B,option(A),T2,K,aa(A,option(A),some(A),X)) = T2 ) ) ).

% map_upd_triv
tff(fact_5790_map__upd__eqD1,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A2: A,X: B,N: fun(A,option(B)),Y: B] :
      ( ( fun_upd(A,option(B),M,A2,aa(B,option(B),some(B),X)) = fun_upd(A,option(B),N,A2,aa(B,option(B),some(B),Y)) )
     => ( X = Y ) ) ).

% map_upd_eqD1
tff(fact_5791_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,T2: fun(A,option(B)),K: A,X: B] :
      ~ ! [X4: A] : aa(A,option(B),fun_upd(A,option(B),T2,K,aa(B,option(B),some(B),X)),X4) = none(B) ).

% map_upd_nonempty
tff(fact_5792_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa(num,num,bit1,M)) = case_nat(option(num),none(num),aTP_Lamp_aau(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_5793_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_aaq(num,option(num)),bit_take_bit_num(pred_numeral(R2),M)) ).

% take_bit_num_simps(6)
tff(fact_5794_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R2),M))) ).

% take_bit_num_simps(7)
tff(fact_5795_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_5796_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),pred_numeral(K))) ) ).

% less_Suc_numeral
tff(fact_5797_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),pred_numeral(K)),N)) ) ).

% less_numeral_Suc
tff(fact_5798_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),pred_numeral(K)),N)) ) ).

% le_numeral_Suc
tff(fact_5799_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),pred_numeral(K))) ) ).

% le_Suc_numeral
tff(fact_5800_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A)] : case_nat(A,F1,F22,zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_5801_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X22: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X22)) = aa(nat,A,F22,X22) ).

% old.nat.simps(5)
tff(fact_5802_nat_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(nat,A),Nat: nat] : aa(A,B,H,case_nat(A,F1,F22,Nat)) = case_nat(B,aa(A,B,H,F1),aa(fun(nat,A),fun(nat,B),aTP_Lamp_aav(fun(A,B),fun(fun(nat,A),fun(nat,B)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_5803_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,N: nat,X: A,F2: fun(nat,A)] :
      ( ( ( N = zero_zero(nat) )
       => ( case_nat(A,X,F2,N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( case_nat(A,X,F2,N) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% Nitpick.case_nat_unfold
tff(fact_5804_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_5805_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : bit_take_bit_num(N,one2) = case_nat(option(num),none(num),aTP_Lamp_aaw(nat,option(num)),N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_5806_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa(num,num,bit0,M)) = case_nat(option(num),none(num),aTP_Lamp_aax(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_5807_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_abb(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_5808_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = none(num) ) )
      & ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)) != zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)))) ) ) ) ).

% take_bit_num_def
tff(fact_5809_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K: A,V2: B] : graph(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V2))) = 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),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V2)),graph(A,B,fun_upd(A,option(B),M,K,none(B)))) ).

% graph_map_upd
tff(fact_5810_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_aat(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_5811_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> pp(case_nat(bool,fTrue,aTP_Lamp_abc(nat,bool),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_5812_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> pp(case_nat(bool,fFalse,aTP_Lamp_abd(nat,bool),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_5813_in__graphD,axiom,
    ! [A: $tType,B: $tType,K: A,V2: B,M: fun(A,option(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V2)),graph(A,B,M)))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V2) ) ) ).

% in_graphD
tff(fact_5814_in__graphI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V2: A] :
      ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V2) )
     => pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V2)),graph(B,A,M))) ) ).

% in_graphI
tff(fact_5815_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_5816_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(case_nat(bool,fFalse,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_5817_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_abe(nat,fun(nat,nat),N),M) ).

% max_Suc1
tff(fact_5818_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_abf(nat,fun(nat,nat),N),M) ).

% max_Suc2
tff(fact_5819_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_5820_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M)),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_5821_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),one_one(nat)))
     => ( num_of_nat(N) = one2 ) ) ).

% num_of_nat_One
tff(fact_5822_graph__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_abg(fun(A,option(B)),fun(product_prod(A,B),bool),M)) ).

% graph_def
tff(fact_5823_graph__fun__upd__None,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K: A] : graph(A,B,fun_upd(A,option(B),M,K,none(B))) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(A,fun(product_prod(A,B),bool),aTP_Lamp_abh(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),M),K)) ).

% graph_fun_upd_None
tff(fact_5824_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_co(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ).

% diff_Suc
tff(fact_5825_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ) ) ) ).

% numeral_num_of_nat_unfold
tff(fact_5826_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = aa(num,num,bit0,num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_5827_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P,case_nat(A,F1,F22,Nat)))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ pp(aa(A,bool,P,F1)) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ pp(aa(A,bool,P,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(2)
tff(fact_5828_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P,case_nat(A,F1,F22,Nat)))
    <=> ( ( ( Nat = zero_zero(nat) )
         => pp(aa(A,bool,P,F1)) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => pp(aa(A,bool,P,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(1)
tff(fact_5829_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),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),X),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_5830_restrict__out,axiom,
    ! [A: $tType,B: $tType,X: A,A3: set(A),M: fun(A,option(B))] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(A,option(B),restrict_map(A,B,M,A3),X) = none(B) ) ) ).

% restrict_out
tff(fact_5831_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D6: set(A),X2: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_aat(A,option(B)),D6),X2) = none(B) ).

% restrict_map_empty
tff(fact_5832_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_5833_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,X: A,D6: set(A),M: fun(A,option(B)),Y: option(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
     => ( fun_upd(A,option(B),restrict_map(A,B,M,D6),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y) ) ) ).

% fun_upd_restrict_conv
tff(fact_5834_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,X: A,D6: set(A),M: fun(A,option(B)),Y: option(B)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
       => ( restrict_map(A,B,fun_upd(A,option(B),M,X,Y),D6) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
       => ( restrict_map(A,B,fun_upd(A,option(B),M,X,Y),D6) = restrict_map(A,B,M,D6) ) ) ) ).

% restrict_fun_upd
tff(fact_5835_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,X: A,D6: set(A),M: fun(A,option(B))] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
       => ( fun_upd(A,option(B),restrict_map(A,B,M,D6),X,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D6))
       => ( fun_upd(A,option(B),restrict_map(A,B,M,D6),X,none(B)) = restrict_map(A,B,M,D6) ) ) ) ).

% fun_upd_None_restrict
tff(fact_5836_restrict__map__def,axiom,
    ! [A: $tType,B: $tType,A3: set(A),M: fun(A,option(B)),X2: A] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
       => ( aa(A,option(B),restrict_map(A,B,M,A3),X2) = aa(A,option(B),M,X2) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
       => ( aa(A,option(B),restrict_map(A,B,M,A3),X2) = none(B) ) ) ) ).

% restrict_map_def
tff(fact_5837_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K: A,V2: B,M: fun(A,option(B)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V2)),graph(A,B,restrict_map(A,B,M,A3))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),K),A3)) ) ).

% graph_restrictD(1)
tff(fact_5838_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_co(nat,nat),Nat) ).

% pred_def
tff(fact_5839_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K: A,V2: B,M: fun(A,option(B)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V2)),graph(A,B,restrict_map(A,B,M,A3))))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V2) ) ) ).

% graph_restrictD(2)
tff(fact_5840_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),D6: set(A),X: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M,D6),X,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),X,Y) ).

% fun_upd_restrict
tff(fact_5841_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),X: A] : restrict_map(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) = fun_upd(A,option(B),F2,X,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_5842_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),D6: set(A),M: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D6))
       => ( restrict_map(A,B,map_upds(A,B,M,Xs,Ys),D6) = map_upds(A,B,restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D6),aa(list(A),set(A),set2(A),Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_5843_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),M,A2) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),M,A2,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),ran(B,A,M)) ) ) ).

% ran_map_upd
tff(fact_5844_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_abi(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys)) ).

% set_zip
tff(fact_5845_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% min.bounded_iff
tff(fact_5846_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_5847_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_5848_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),Y)) ) ) ) ).

% min_less_iff_conj
tff(fact_5849_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_5850_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb3
tff(fact_5851_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),top_top(A)) = X ) ).

% min_top2
tff(fact_5852_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),X) = X ) ).

% min_top
tff(fact_5853_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_5854_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),X) = bot_bot(A) ) ).

% min_bot
tff(fact_5855_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = X ) ).

% max_min_same(1)
tff(fact_5856_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),X) = X ) ).

% max_min_same(2)
tff(fact_5857_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Y) = Y ) ).

% max_min_same(3)
tff(fact_5858_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = Y ) ).

% max_min_same(4)
tff(fact_5859_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)) ).

% min_Suc_Suc
tff(fact_5860_min__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% min_0L
tff(fact_5861_min__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_5862_map__upds__apply__nontin,axiom,
    ! [B: $tType,A: $tType,X: A,Xs: list(A),F2: fun(A,option(B)),Ys: list(B)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(A,option(B),map_upds(A,B,F2,Xs,Ys),X) = aa(A,option(B),F2,X) ) ) ).

% map_upds_apply_nontin
tff(fact_5863_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) ) ) ) ).

% min_number_of(1)
tff(fact_5864_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_5865_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_5866_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_5867_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_5868_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_abj(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_5869_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) ) ) ) ).

% min_number_of(4)
tff(fact_5870_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) ) ) ) ).

% min_number_of(3)
tff(fact_5871_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) ) ) ) ).

% min_number_of(2)
tff(fact_5872_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,M: fun(A,option(B)),Ys: list(B),Y: B] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I))
     => ( map_upds(A,B,M,Xs,aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),I),Y)) = map_upds(A,B,M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_5873_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] : zip(A,B,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),I),X),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),J),Y)) = aa(product_prod(A,B),list(product_prod(A,B)),aa(nat,fun(product_prod(A,B),list(product_prod(A,B))),replicate(product_prod(A,B)),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_replicate
tff(fact_5874_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A2: A,As: list(A),M: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),As)))
     => ( map_upds(A,B,fun_upd(A,option(B),M,A2,aa(B,option(B),some(B),B2)),As,Bs) = fun_upd(A,option(B),map_upds(A,B,M,As,Bs),A2,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_5875_length__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),zip(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_zip
tff(fact_5876_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_5877_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% min_add_distrib_right
tff(fact_5878_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2)) ) ) ).

% min.strict_coboundedI2
tff(fact_5879_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2)) ) ) ).

% min.strict_coboundedI1
tff(fact_5880_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_5881_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% min.strict_boundedE
tff(fact_5882_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z)) ) ) ) ).

% min_less_iff_disj
tff(fact_5883_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),I)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),I) ).

% min_diff
tff(fact_5884_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_5885_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)) ).

% nat_mult_min_left
tff(fact_5886_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)) ).

% nat_mult_min_right
tff(fact_5887_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_5888_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_5889_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_5890_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_5891_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ) ).

% min_def
tff(fact_5892_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).

% min_absorb1
tff(fact_5893_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_5894_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ) ).

% min_le_iff_disj
tff(fact_5895_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2)) ) ) ).

% min.coboundedI2
tff(fact_5896_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2)) ) ) ).

% min.coboundedI1
tff(fact_5897_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_5898_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb_iff1
tff(fact_5899_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2)) ) ).

% min.cobounded2
tff(fact_5900_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2)) ) ).

% min.cobounded1
tff(fact_5901_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.order_iff
tff(fact_5902_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))) ) ) ) ).

% min.boundedI
tff(fact_5903_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% min.boundedE
tff(fact_5904_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% min.orderI
tff(fact_5905_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.orderE
tff(fact_5906_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D2))) ) ) ) ).

% min.mono
tff(fact_5907_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Xa) = X2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Xa) = Xa ) ) ) ) ).

% min_def_raw
tff(fact_5908_ranI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),M,A2) = aa(A,option(A),some(A),B2) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ran(B,A,M))) ) ).

% ranI
tff(fact_5909_ran__restrictD,axiom,
    ! [B: $tType,A: $tType,Y: A,M: fun(B,option(A)),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),ran(B,A,restrict_map(B,A,M,A3))))
     => ? [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
          & ( aa(B,option(A),M,X4) = aa(A,option(A),some(A),Y) ) ) ) ).

% ran_restrictD
tff(fact_5910_ran__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : ran(A,B,M) = aa(fun(B,bool),set(B),collect(B),aTP_Lamp_abk(fun(A,option(B)),fun(B,bool),M)) ).

% ran_def
tff(fact_5911_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) ) ) ) ).

% min_mult_distrib_right
tff(fact_5912_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) ) ) ) ).

% max_mult_distrib_right
tff(fact_5913_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) ) ) ) ).

% min_mult_distrib_left
tff(fact_5914_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) ) ) ) ).

% max_mult_distrib_left
tff(fact_5915_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,X,P2)),divide_divide(A,Y,P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,X,P2)),divide_divide(A,Y,P2)) ) ) ) ) ).

% min_divide_distrib_right
tff(fact_5916_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,X,P2)),divide_divide(A,Y,P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,X,P2)),divide_divide(A,Y,P2)) ) ) ) ) ).

% max_divide_distrib_right
tff(fact_5917_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_abl(nat,fun(nat,nat),N),M) ).

% min_Suc2
tff(fact_5918_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_abm(nat,fun(nat,nat),N),M) ).

% min_Suc1
tff(fact_5919_Inf__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( ( S2 = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S2)) = X ) )
            & ( ( S2 != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,complete_Inf_Inf(A),S2)) ) ) ) ) ) ).

% Inf_insert_finite
tff(fact_5920_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),X: B,Y: A,Z: A] :
      ( ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y) )
     => ( inj_on(B,option(A),M,dom(B,A,M))
       => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),ran(B,A,M)))
         => ( ran(B,A,fun_upd(B,option(A),M,X,aa(A,option(A),some(A),Z))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),ran(B,A,M)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),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_5921_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_abn(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Min.eq_fold'
tff(fact_5922_map__upds__fold__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,M,Ks,Vs) = aa(list(product_prod(A,B)),fun(A,option(B)),aa(fun(A,option(B)),fun(list(product_prod(A,B)),fun(A,option(B))),aa(fun(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),fun(fun(A,option(B)),fun(list(product_prod(A,B)),fun(A,option(B)))),foldl(fun(A,option(B)),product_prod(A,B)),aTP_Lamp_abp(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))))),M),zip(A,B,Ks,Vs)) ).

% map_upds_fold_map_upd
tff(fact_5923_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = X ) ).

% Min_singleton
tff(fact_5924_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( ( dom(A,B,F2) = bot_bot(set(A)) )
    <=> ! [X3: A] : aa(A,option(B),F2,X3) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_5925_fun__upd__None__if__notin__dom,axiom,
    ! [B: $tType,A: $tType,K: A,M: fun(A,option(B))] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),K),dom(A,B,M)))
     => ( fun_upd(A,option(B),M,K,none(B)) = M ) ) ).

% fun_upd_None_if_notin_dom
tff(fact_5926_dom__const,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : dom(A,B,aTP_Lamp_abq(fun(A,B),fun(A,option(B)),F2)) = top_top(set(A)) ).

% dom_const
tff(fact_5927_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_aat(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_5928_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X3)) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_5929_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)))
            <=> ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X3)) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_5930_Min__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [A3: set(B),C2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( A3 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_uv(A,fun(B,A),C2)),A3)) = C2 ) ) ) ) ).

% Min_const
tff(fact_5931_finite__graph__iff__finite__dom,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),graph(A,B,M)))
    <=> pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M))) ) ).

% finite_graph_iff_finite_dom
tff(fact_5932_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) ) ) ) ) ).

% Min_insert
tff(fact_5933_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798350308766er_Min(A),S2)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S2)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_5934_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798349783984er_Max(A),S2)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S2)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_5935_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y: option(B),F2: fun(A,option(B)),X: A] :
      ( ( ( Y = none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F2,X,Y)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) )
      & ( ( Y != none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F2,X,Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),dom(A,B,F2)) ) ) ) ).

% dom_fun_upd
tff(fact_5936_insert__dom,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),X: B,Y: A] :
      ( ( aa(B,option(A),F2,X) = aa(A,option(A),some(A),Y) )
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),dom(B,A,F2)) = dom(B,A,F2) ) ) ).

% insert_dom
tff(fact_5937_dom__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abr(fun(A,option(B)),fun(A,bool),M)) ).

% dom_def
tff(fact_5938_domI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),M,A2) = aa(A,option(A),some(A),B2) )
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),dom(B,A,M))) ) ).

% domI
tff(fact_5939_domD,axiom,
    ! [A: $tType,B: $tType,A2: A,M: fun(A,option(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),dom(A,B,M)))
     => ? [B4: B] : aa(A,option(B),M,A2) = aa(B,option(B),some(B),B4) ) ).

% domD
tff(fact_5940_domIff,axiom,
    ! [A: $tType,B: $tType,A2: A,M: fun(A,option(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),dom(A,B,M)))
    <=> ( aa(A,option(B),M,A2) != none(B) ) ) ).

% domIff
tff(fact_5941_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A3)) ) ) ) ).

% Min_in
tff(fact_5942_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A2)) ) ) ) ).

% Min.coboundedI
tff(fact_5943_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ! [Y2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = X ) ) ) ) ) ).

% Min_eqI
tff(fact_5944_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),X)) ) ) ) ).

% Min_le
tff(fact_5945_Inf__fin__Min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).

% Inf_fin_Min
tff(fact_5946_foldl__Nil,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,B)),A2: B] : aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F2),A2),nil(A)) = A2 ).

% foldl_Nil
tff(fact_5947_foldl__cong,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: A,L: list(B),K: list(B),F2: fun(A,fun(B,A)),G: fun(A,fun(B,A))] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A4: A,X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(list(B),set(B),set2(B),L)))
             => ( aa(B,A,aa(A,fun(B,A),F2,A4),X4) = aa(B,A,aa(A,fun(B,A),G,A4),X4) ) )
         => ( aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A2),L) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),G),B2),K) ) ) ) ) ).

% foldl_cong
tff(fact_5948_Min_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(A),A,lattic643756798350308766er_Min(A),A3) ) ) ) ) ).

% Min.in_idem
tff(fact_5949_finite__ran,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,option(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,P2)))
     => pp(aa(set(B),bool,finite_finite2(B),ran(A,B,P2))) ) ).

% finite_ran
tff(fact_5950_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,F2)))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
       => ? [X4: A] : aa(A,option(B),F2,X4) = none(B) ) ) ).

% finite_map_freshness
tff(fact_5951_dom__minus,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),X: B,A3: set(B)] :
      ( ( aa(B,option(A),F2,X) = none(A) )
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F2)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F2)),A3) ) ) ).

% dom_minus
tff(fact_5952_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A4)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3))) ) ) ) ) ).

% Min.boundedI
tff(fact_5953_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)))
             => ! [A10: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A10),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A10)) ) ) ) ) ) ).

% Min.boundedE
tff(fact_5954_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A3) )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X3)) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_5955_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),X))
            <=> ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),X)) ) ) ) ) ) ).

% Min_le_iff
tff(fact_5956_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = M )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X3)) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_5957_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),X))
            <=> ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),X)) ) ) ) ) ) ).

% Min_less_iff
tff(fact_5958_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B4)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_5959_cInf__eq__Min,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = aa(set(A),A,lattic643756798350308766er_Min(A),X6) ) ) ) ) ).

% cInf_eq_Min
tff(fact_5960_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(set(A),A,complete_Inf_Inf(A),A3) ) ) ) ) ).

% Min_Inf
tff(fact_5961_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Min.infinite
tff(fact_5962_finite__set__of__finite__maps,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B3))
       => pp(aa(set(fun(A,option(B))),bool,finite_finite2(fun(A,option(B))),aa(fun(fun(A,option(B)),bool),set(fun(A,option(B))),collect(fun(A,option(B))),aa(set(B),fun(fun(A,option(B)),bool),aTP_Lamp_abs(set(A),fun(set(B),fun(fun(A,option(B)),bool)),A3),B3)))) ) ) ).

% finite_set_of_finite_maps
tff(fact_5963_graph__eq__to__snd__dom,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(set(A),set(product_prod(A,B)),image2(A,product_prod(A,B),aTP_Lamp_abt(fun(A,option(B)),fun(A,product_prod(A,B)),M)),dom(A,B,M)) ).

% graph_eq_to_snd_dom
tff(fact_5964_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B3)),aa(set(A),A,lattic643756798350308766er_Min(A),A3))) ) ) ) ) ).

% Min.subset_imp
tff(fact_5965_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M4: set(A),N4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M4),N4))
         => ( ( M4 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),N4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N4)),aa(set(A),A,lattic643756798350308766er_Min(A),M4))) ) ) ) ) ).

% Min_antimono
tff(fact_5966_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N4: set(A)] :
          ( ! [X4: A,Y2: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_min(A),X4),Y2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H,X4)),aa(A,A,H,Y2))
         => ( pp(aa(set(A),bool,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_5967_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( B3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B3)),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(A),A,lattic643756798350308766er_Min(A),A3) ) ) ) ) ) ).

% Min.subset
tff(fact_5968_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X4: A,Y2: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X4),Y2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A3)) ) ) ) ) ).

% Min.closed
tff(fact_5969_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( A3 != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_5970_mono__Min__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( pp(aa(set(A),bool,finite_finite2(A),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(A,B,F2,aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_5971_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => ( ( B3 != 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)),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),aa(set(A),A,lattic643756798350308766er_Min(A),B3)) ) ) ) ) ) ) ).

% Min.union
tff(fact_5972_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,A,ord_min(A),X,A3) ) ) ) ).

% Min.eq_fold
tff(fact_5973_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(fun(A,option(B)),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M)))
     => ( pp(aa(fun(A,option(B)),bool,P,aTP_Lamp_aat(A,option(B))))
       => ( ! [K2: A,V3: B,M5: fun(A,option(B))] :
              ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M5)))
             => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),K2),dom(A,B,M5)))
               => ( pp(aa(fun(A,option(B)),bool,P,M5))
                 => pp(aa(fun(A,option(B)),bool,P,fun_upd(A,option(B),M5,K2,aa(B,option(B),some(B),V3)))) ) ) )
         => pp(aa(fun(A,option(B)),bool,P,M)) ) ) ) ).

% finite_Map_induct
tff(fact_5974_Min__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S2: set(B),F2: fun(B,A),K: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( S2 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_uy(fun(B,A),fun(A,fun(B,A)),F2),K)),S2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image2(B,A,F2),S2))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_5975_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Min.remove
tff(fact_5976_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = 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),X),A3)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) != 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),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Min.insert_remove
tff(fact_5977_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),X: A] :
      ( ( dom(A,B,F2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) )
    <=> ? [V6: B] : F2 = fun_upd(A,option(B),aTP_Lamp_aat(A,option(B)),X,aa(B,option(B),some(B),V6)) ) ).

% dom_eq_singleton_conv
tff(fact_5978_card__Min__le__sum,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(A),set(nat),image2(A,nat,F2),A3)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3))) ) ).

% card_Min_le_sum
tff(fact_5979_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_rx(A,fun(A,bool))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_5980_f__arg__min__list__f,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( aa(A,B,F2,arg_min_list(A,B,F2,Xs)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),Xs))) ) ) ) ).

% f_arg_min_list_f
tff(fact_5981_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),aa(num,num,bit0,N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_abu(num,option(num)),bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_5982_linorder_OMax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Max(A,Less_eq) = lattices_Max(A,Less_eq) ).

% linorder.Max.cong
tff(fact_5983_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_5984_arg__min__list__in,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B)] :
          ( ( Xs != nil(A) )
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),arg_min_list(A,B,F2,Xs)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% arg_min_list_in
tff(fact_5985_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit0,M),one2) = aa(num,option(num),some(num),aa(num,num,bit0,M)) ).

% and_not_num.simps(4)
tff(fact_5986_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_and_not_num(one2,aa(num,num,bit0,N)) = aa(num,option(num),some(num),one2) ).

% and_not_num.simps(2)
tff(fact_5987_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_and_not_num(one2,aa(num,num,bit1,N)) = none(num) ).

% and_not_num.simps(3)
tff(fact_5988_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit1,M),one2) = aa(num,option(num),some(num),aa(num,num,bit0,M)) ).

% and_not_num.simps(7)
tff(fact_5989_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa2 = aa(num,num,bit0,N2)
             => ( Y != aa(num,option(num),some(num),one2) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa2 = aa(num,num,bit1,N2)
               => ( Y != none(num) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa(num,num,bit0,M5) )
                 => ( ( Xa2 = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa(num,num,bit0,N2) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit1,N2) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit1,M5) )
                       => ( ( Xa2 = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa(num,num,bit0,N2) )
                             => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_abu(num,option(num)),bit_and_not_num(M5,N2)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit1,N2) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_5990_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_un2480387367778600638or_num(aa(num,num,bit1,M),aa(num,num,bit0,N)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M,N))) ).

% xor_num.simps(8)
tff(fact_5991_map__option__eq__Some,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xo: option(B),Y: A] :
      ( ( aa(option(B),option(A),map_option(B,A,F2),Xo) = aa(A,option(A),some(A),Y) )
    <=> ? [Z2: B] :
          ( ( Xo = aa(B,option(B),some(B),Z2) )
          & ( aa(B,A,F2,Z2) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_5992_option_Omap__disc__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: option(A)] :
      ( ( aa(option(A),option(B),map_option(A,B,F2),A2) = none(B) )
    <=> ( A2 = none(A) ) ) ).

% option.map_disc_iff
tff(fact_5993_map__option__is__None,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Opt: option(B)] :
      ( ( aa(option(B),option(A),map_option(B,A,F2),Opt) = none(A) )
    <=> ( Opt = none(B) ) ) ).

% map_option_is_None
tff(fact_5994_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: option(B)] :
      ( ( none(A) = aa(option(B),option(A),map_option(B,A,F2),X) )
    <=> ( X = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_5995_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,B),X2: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F2)),aTP_Lamp_abv(A,option(C))),X2) = none(B) ).

% map_option_o_empty
tff(fact_5996_case__map__option,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: A,H: fun(B,A),F2: fun(C,B),X: option(C)] : case_option(A,B,G,H,aa(option(C),option(B),map_option(C,B,F2),X)) = case_option(A,C,G,aa(fun(C,B),fun(C,A),comp(B,A,C,H),F2),X) ).

% case_map_option
tff(fact_5997_map__option__o__map__upd,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,B),M: fun(A,option(C)),A2: A,B2: C] : aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F2)),fun_upd(A,option(C),M,A2,aa(C,option(C),some(C),B2))) = fun_upd(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F2)),M),A2,aa(B,option(B),some(B),aa(C,B,F2,B2))) ).

% map_option_o_map_upd
tff(fact_5998_map__option_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,C),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),option(C)),comp(option(B),option(C),option(A),map_option(B,C,F2)),map_option(A,B,G)) = map_option(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,F2),G)) ).

% map_option.comp
tff(fact_5999_option_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F2: fun(A,B),V2: option(A)] : aa(option(B),option(C),map_option(B,C,G),aa(option(A),option(B),map_option(A,B,F2),V2)) = aa(option(A),option(C),map_option(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,G),F2)),V2) ).

% option.map_comp
tff(fact_6000_map__option_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(B,C),G: fun(A,B),Option: option(A)] : aa(option(B),option(C),map_option(B,C,F2),aa(option(A),option(B),map_option(A,B,G),Option)) = aa(option(A),option(C),map_option(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,F2),G)),Option) ).

% map_option.compositionality
tff(fact_6001_option_Omap__ident,axiom,
    ! [A: $tType,T2: option(A)] : aa(option(A),option(A),map_option(A,A,aTP_Lamp_ki(A,A)),T2) = T2 ).

% option.map_ident
tff(fact_6002_option_Omap__sel,axiom,
    ! [B: $tType,A: $tType,A2: option(A),F2: fun(A,B)] :
      ( ( A2 != none(A) )
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),map_option(A,B,F2),A2)) = aa(A,B,F2,aa(option(A),A,the2(A),A2)) ) ) ).

% option.map_sel
tff(fact_6003_option_Osimps_I9_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X22: A] : aa(option(A),option(B),map_option(A,B,F2),aa(A,option(A),some(A),X22)) = aa(B,option(B),some(B),aa(A,B,F2,X22)) ).

% option.simps(9)
tff(fact_6004_map__option__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( X = Y )
     => ( ! [A4: A] :
            ( ( Y = aa(A,option(A),some(A),A4) )
           => ( aa(A,B,F2,A4) = aa(A,B,G,A4) ) )
       => ( aa(option(A),option(B),map_option(A,B,F2),X) = aa(option(A),option(B),map_option(A,B,G),Y) ) ) ) ).

% map_option_cong
tff(fact_6005_option_Osimps_I8_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B)] : aa(option(A),option(B),map_option(A,B,F2),none(A)) = none(B) ).

% option.simps(8)
tff(fact_6006_xor__num_Osimps_I1_J,axiom,
    bit_un2480387367778600638or_num(one2,one2) = none(num) ).

% xor_num.simps(1)
tff(fact_6007_option_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,nat),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),comp(option(B),nat,option(A),size_option(B,F2)),map_option(A,B,G)) = size_option(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F2),G)) ).

% option.size_gen_o_map
tff(fact_6008_option_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(option(A),option(B),map_option(A,B,F2),top_top(set(option(A)))) ) ).

% option.inj_map
tff(fact_6009_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un2480387367778600638or_num(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( X = one2 )
           => ! [N2: num] :
                ( ( Xa2 = aa(num,num,bit0,N2) )
               => ( Y != aa(num,option(num),some(num),aa(num,num,bit1,N2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa2 = aa(num,num,bit1,N2) )
                 => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,N2)) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa(num,num,bit0,M5) )
                 => ( ( Xa2 = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit1,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa(num,num,bit0,N2) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M5,N2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit1,N2) )
                         => ( Y != aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M5,N2))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit1,M5) )
                       => ( ( Xa2 = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa(num,num,bit0,N2) )
                             => ( Y != aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M5,N2))) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit1,N2) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M5,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_6010_map__option__case,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Y: option(B)] : aa(option(B),option(A),map_option(B,A,F2),Y) = case_option(option(A),B,none(A),aTP_Lamp_abw(fun(B,A),fun(B,option(A)),F2),Y) ).

% map_option_case
tff(fact_6011_xor__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num,Q4: num] :
          ( ( bit_un2480387367778600638or_num(M,N) = aa(num,option(num),some(num),Q4) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q4) ) ) ) ).

% xor_num_eq_Some_iff
tff(fact_6012_xor__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_un2480387367778600638or_num(one2,aa(num,num,bit0,N)) = aa(num,option(num),some(num),aa(num,num,bit1,N)) ).

% xor_num.simps(2)
tff(fact_6013_xor__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_un2480387367778600638or_num(one2,aa(num,num,bit1,N)) = aa(num,option(num),some(num),aa(num,num,bit0,N)) ).

% xor_num.simps(3)
tff(fact_6014_xor__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_un2480387367778600638or_num(aa(num,num,bit0,M),one2) = aa(num,option(num),some(num),aa(num,num,bit1,M)) ).

% xor_num.simps(4)
tff(fact_6015_xor__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_un2480387367778600638or_num(aa(num,num,bit1,M),one2) = aa(num,option(num),some(num),aa(num,num,bit0,M)) ).

% xor_num.simps(7)
tff(fact_6016_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] :
          ( ( bit_un2480387367778600638or_num(M,N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_6017_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un2480387367778600638or_num(M,N)) ) ).

% numeral_xor_num
tff(fact_6018_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : bit_un2480387367778600638or_num(aa(num,num,bit0,M),aa(num,num,bit1,N)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M,N))) ).

% xor_num.simps(6)
tff(fact_6019_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != aa(num,option(num),some(num),one2) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa2 = aa(num,num,bit0,N2)
             => ( Y != none(num) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa2 = aa(num,num,bit1,N2)
               => ( Y != aa(num,option(num),some(num),one2) ) ) )
           => ( ( ? [M5: num] : X = aa(num,num,bit0,M5)
               => ( ( Xa2 = one2 )
                 => ( Y != none(num) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N2: num] :
                        ( ( Xa2 = aa(num,num,bit0,N2) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit1,N2) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N2)) ) ) )
                 => ( ( ? [M5: num] : X = aa(num,num,bit1,M5)
                     => ( ( Xa2 = one2 )
                       => ( Y != aa(num,option(num),some(num),one2) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N2: num] :
                              ( ( Xa2 = aa(num,num,bit0,N2) )
                             => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N2)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit1,N2) )
                               => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_abu(num,option(num)),bit_un7362597486090784418nd_num(M5,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_6020_min__list__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( min_list(A,Xs) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% min_list_Min
tff(fact_6021_and__num_Osimps_I1_J,axiom,
    bit_un7362597486090784418nd_num(one2,one2) = aa(num,option(num),some(num),one2) ).

% and_num.simps(1)
tff(fact_6022_and__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_un7362597486090784418nd_num(aa(num,num,bit1,M),one2) = aa(num,option(num),some(num),one2) ).

% and_num.simps(7)
tff(fact_6023_and__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_un7362597486090784418nd_num(one2,aa(num,num,bit1,N)) = aa(num,option(num),some(num),one2) ).

% and_num.simps(3)
tff(fact_6024_and__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_un7362597486090784418nd_num(aa(num,num,bit0,M),one2) = none(num) ).

% and_num.simps(4)
tff(fact_6025_and__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_un7362597486090784418nd_num(one2,aa(num,num,bit0,N)) = none(num) ).

% and_num.simps(2)
tff(fact_6026_and__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num,Q4: num] :
          ( ( bit_un7362597486090784418nd_num(M,N) = aa(num,option(num),some(num),Q4) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q4) ) ) ) ).

% and_num_eq_Some_iff
tff(fact_6027_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] :
          ( ( bit_un7362597486090784418nd_num(M,N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_6028_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un7362597486090784418nd_num(M,N)) ) ).

% numeral_and_num
tff(fact_6029_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_un7362597486090784418nd_num(aa(num,num,bit1,M),aa(num,num,bit1,N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_abu(num,option(num)),bit_un7362597486090784418nd_num(M,N)) ).

% and_num.simps(9)
tff(fact_6030_option_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,C),F2: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),C),comp(option(B),C,option(A),rec_option(C,B,G,Ga)),map_option(A,B,F2)) = rec_option(C,A,G,aa(fun(A,B),fun(A,C),aTP_Lamp_yd(fun(B,C),fun(fun(A,B),fun(A,C)),Ga),F2)) ).

% option.rec_o_map
tff(fact_6031_minus__fold__remove,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3) = finite_fold(A,set(A),remove(A),B3,A3) ) ) ).

% minus_fold_remove
tff(fact_6032_member__remove,axiom,
    ! [A: $tType,X: A,Y: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Y),A3)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_6033_option_Osimps_I7_J,axiom,
    ! [C: $tType,A: $tType,F1: C,F22: fun(A,C),X22: A] : aa(option(A),C,rec_option(C,A,F1,F22),aa(A,option(A),some(A),X22)) = aa(A,C,F22,X22) ).

% option.simps(7)
tff(fact_6034_option_Osimps_I6_J,axiom,
    ! [A: $tType,C: $tType,F1: C,F22: fun(A,C)] : aa(option(A),C,rec_option(C,A,F1,F22),none(A)) = F1 ).

% option.simps(6)
tff(fact_6035_remove__code_I1_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,X),Xs)) ).

% remove_code(1)
tff(fact_6036_remove__def,axiom,
    ! [A: $tType,X: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ).

% remove_def
tff(fact_6037_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I6)) = aa(fun(A,bool),set(A),collect(A),aa(set(nat),fun(A,bool),aTP_Lamp_abx(list(A),fun(set(nat),fun(A,bool)),Xs),I6)) ).

% set_nths
tff(fact_6038_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = inc(num_of_nat(N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = one2 ) ) ) ).

% num_of_nat.simps(2)
tff(fact_6039_nths__nil,axiom,
    ! [A: $tType,A3: set(nat)] : nths(A,nil(A),A3) = nil(A) ).

% nths_nil
tff(fact_6040_nths__empty,axiom,
    ! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).

% nths_empty
tff(fact_6041_notin__set__nthsI,axiom,
    ! [A: $tType,X: A,Xs: list(A),I6: set(nat)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),nths(A,Xs,I6)))) ) ).

% notin_set_nthsI
tff(fact_6042_in__set__nthsD,axiom,
    ! [A: $tType,X: A,Xs: list(A),I6: set(nat)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),nths(A,Xs,I6))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_nthsD
tff(fact_6043_distinct__nthsI,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( distinct(A,Xs)
     => distinct(A,nths(A,Xs,I6)) ) ).

% distinct_nthsI
tff(fact_6044_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I6))),aa(list(A),set(A),set2(A),Xs))) ).

% set_nths_subset
tff(fact_6045_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),I2),I6)) )
     => ( nths(A,Xs,I6) = Xs ) ) ).

% nths_all
tff(fact_6046_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I6)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_aby(list(A),fun(set(nat),fun(nat,bool)),Xs),I6))) ).

% length_nths
tff(fact_6047_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = aa(fun(real,bool),set(real),collect(real),aTP_Lamp_abz(real,bool)) ).

% Rats_eq_int_div_nat
tff(fact_6048_listrel__def,axiom,
    ! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : listrel(A,B,X2) = aa(fun(product_prod(list(A),list(B)),bool),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),bool)),fun(product_prod(list(A),list(B)),bool),product_case_prod(list(A),list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),X2)))) ).

% listrel_def
tff(fact_6049_Rats__infinite,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ~ pp(aa(set(A),bool,finite_finite2(A),field_char_0_Rats(A))) ) ).

% Rats_infinite
tff(fact_6050_listrelp_ONil,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),nil(A)),nil(B))) ).

% listrelp.Nil
tff(fact_6051_Rats__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),field_char_0_Rats(A))) ) ).

% Rats_0
tff(fact_6052_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X4: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),field_char_0_Rats(real)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),X)) ) ).

% Rats_no_bot_less
tff(fact_6053_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
     => ? [X4: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X4),field_char_0_Rats(real)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),X4))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X4),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_6054_Ints__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ring_1_Ints(A)),field_char_0_Rats(A))) ) ).

% Ints_subset_Rats
tff(fact_6055_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X2: list(A),Xa: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),X2),Xa))
    <=> pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),X2),Xa)),listrel(A,B,R2))) ) ).

% listrelp_listrel_eq
tff(fact_6056_pos__deriv__imp__strict__mono,axiom,
    ! [F2: fun(real,real),F9: fun(real,real)] :
      ( ! [X4: real] : has_field_derivative(real,F2,aa(real,real,F9,X4),topolo174197925503356063within(real,X4,top_top(set(real))))
     => ( ! [X4: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,F9,X4)))
       => order_strict_mono(real,real,F2) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_6057_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F2,X),finite_fold(A,B,F2,Z,A3)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_6058_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
       => ( 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.comp_fun_idem_on
tff(fact_6059_comp__fun__idem__on_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => finite4664212375090638736ute_on(A,B,S2,F2) ) ).

% comp_fun_idem_on.axioms(1)
tff(fact_6060_strict__mono__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
         => pp(aa(fun(A,B),bool,order_mono(A,B),F2)) ) ) ).

% strict_mono_mono
tff(fact_6061_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% strict_mono_less
tff(fact_6062_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
        <=> ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y4))) ) ) ) ).

% strict_mono_def
tff(fact_6063_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,Y2))) )
         => order_strict_mono(A,B,F2) ) ) ).

% strict_monoI
tff(fact_6064_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y))) ) ) ) ).

% strict_monoD
tff(fact_6065_strict__mono__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( ( aa(A,B,F2,X) = aa(A,B,F2,Y) )
          <=> ( X = Y ) ) ) ) ).

% strict_mono_eq
tff(fact_6066_comp__fun__idem__on_Ofun__left__idem,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
       => ( aa(B,B,aa(A,fun(B,B),F2,X),aa(B,B,aa(A,fun(B,B),F2,X),Z)) = aa(B,B,aa(A,fun(B,B),F2,X),Z) ) ) ) ).

% comp_fun_idem_on.fun_left_idem
tff(fact_6067_strict__mono__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [R2: fun(A,B),M: A,N: A] :
          ( order_strict_mono(A,B,R2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),N))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,R2,M)),aa(A,B,R2,N))) ) ) ) ).

% strict_mono_leD
tff(fact_6068_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% strict_mono_less_eq
tff(fact_6069_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F2: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G),top_top(set(C)))),S2))
       => 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_6070_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,X),Z),A3) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_6071_positive__rat,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(rat,bool,positive,aa(int,rat,aa(int,fun(int,rat),fract,A2),B2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2))) ) ).

% positive_rat
tff(fact_6072_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_rx(A,fun(A,bool))) = ord_min(A) ) ) ).

% dual_max
tff(fact_6073_strict__mono__imp__increasing,axiom,
    ! [F2: fun(nat,nat),N: nat] :
      ( order_strict_mono(nat,nat,F2)
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,F2,N))) ) ).

% strict_mono_imp_increasing
tff(fact_6074_infinite__enumerate,axiom,
    ! [S2: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => ? [R3: fun(nat,nat)] :
          ( order_strict_mono(nat,nat,R3)
          & ! [N5: nat] : pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,R3,N5)),S2)) ) ) ).

% infinite_enumerate
tff(fact_6075_ord_Omax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : max(A,Less_eq) = max(A,Less_eq) ).

% ord.max.cong
tff(fact_6076_ord_Omax__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A2: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A2),B2) = B2 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A2),B2) = A2 ) ) ) ).

% ord.max_def
tff(fact_6077_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_strict_mono(nat,A,F2)
        <=> ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F2,N3)),aa(nat,A,F2,aa(nat,nat,suc,N3)))) ) ) ).

% strict_mono_Suc_iff
tff(fact_6078_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
    <=> pp(aa(rat,bool,positive,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Y),X))) ) ).

% less_rat_def
tff(fact_6079_summable__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),aa(set(nat),set(nat),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_aca(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2))
            <=> summable(A,F2) ) ) ) ) ).

% summable_mono_reindex
tff(fact_6080_sums__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A),C2: A] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),aa(set(nat),set(nat),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_aca(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2),C2)
            <=> sums(A,F2,C2) ) ) ) ) ).

% sums_mono_reindex
tff(fact_6081_suminf__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N2: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),aa(set(nat),set(nat),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_acb(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2)) = suminf(A,F2) ) ) ) ) ).

% suminf_mono_reindex
tff(fact_6082_increasing__Bseq__subseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,nat)] :
          ( ! [X4: nat,Y2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Y2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,X4))),real_V7770717601297561774m_norm(A,aa(nat,A,F2,Y2)))) )
         => ( order_strict_mono(nat,nat,G)
           => ( bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_acc(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_6083_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa(rat,bool,positive,X))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,X))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_6084_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(set(nat),set(A),image2(nat,A,nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),L),Xs)) ) ) ).

% nth_image
tff(fact_6085_take__take,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),M)),Xs) ).

% take_take
tff(fact_6086_take0,axiom,
    ! [A: $tType,X2: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),zero_zero(nat)),X2) = nil(A) ).

% take0
tff(fact_6087_take__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) = nil(A) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_6088_take__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_6089_take__all__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) = Xs )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% take_all_iff
tff(fact_6090_take__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) = Xs ) ) ).

% take_all
tff(fact_6091_nth__take,axiom,
    ! [A: $tType,I: nat,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),I) = aa(nat,A,nth(A,Xs),I) ) ) ).

% nth_take
tff(fact_6092_take__update__cancel,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),M),Y)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) ) ) ).

% take_update_cancel
tff(fact_6093_length__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_take
tff(fact_6094_nths__upt__eq__take,axiom,
    ! [A: $tType,L: list(A),N: nat] : nths(A,L,set_ord_lessThan(nat,N)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),L) ).

% nths_upt_eq_take
tff(fact_6095_take__replicate,axiom,
    ! [A: $tType,I: nat,K: nat,X: A] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),K)),X) ).

% take_replicate
tff(fact_6096_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xs: list(A),Ys: list(B)] : dom(A,B,map_upds(A,B,M,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(B),nat,size_size(list(B)),Ys)),Xs))),dom(A,B,M)) ).

% dom_map_upds
tff(fact_6097_in__set__takeD,axiom,
    ! [A: $tType,X: A,N: nat,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_takeD
tff(fact_6098_take__zip,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(nat,fun(list(product_prod(A,B)),list(product_prod(A,B))),take(product_prod(A,B)),N),zip(A,B,Xs,Ys)) = zip(A,B,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Ys)) ).

% take_zip
tff(fact_6099_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),zero_zero(nat)),Xs) = nil(A) ).

% take_0
tff(fact_6100_take__update__swap,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat,X: A] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M),Xs)),N),X) ).

% take_update_swap
tff(fact_6101_take__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ! [I2: nat] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),Ys)
     => ( Xs = Ys ) ) ).

% take_equalityI
tff(fact_6102_distinct__take,axiom,
    ! [A: $tType,Xs: list(A),I: nat] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)) ) ).

% distinct_take
tff(fact_6103_take__Nil,axiom,
    ! [A: $tType,N: nat] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),nil(A)) = nil(A) ).

% take_Nil
tff(fact_6104_set__take__subset,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_take_subset
tff(fact_6105_set__take__subset__set__take,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M),Xs))),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)))) ) ).

% set_take_subset_set_take
tff(fact_6106_zip__obtain__same__length,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(product_prod(A,B)),bool)] :
      ( ! [Zs2: list(A),Ws: list(B),N2: nat] :
          ( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(B),nat,size_size(list(B)),Ws) )
         => ( ( N2 = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) )
           => ( ( Zs2 = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N2),Xs) )
             => ( ( Ws = aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N2),Ys) )
               => pp(aa(list(product_prod(A,B)),bool,P,zip(A,B,Zs2,Ws))) ) ) ) )
     => pp(aa(list(product_prod(A,B)),bool,P,zip(A,B,Xs,Ys))) ) ).

% zip_obtain_same_length
tff(fact_6107_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys)))
       => ( ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K))
             => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
         => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),K),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),K),Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_6108_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,X: A,Ys: list(B),Xs: list(A),F2: fun(A,option(B)),Y: B] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(B),nat,size_size(list(B)),Ys)),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F2,X,aa(B,option(B),some(B),Y)),Xs,Ys) = map_upds(A,B,F2,Xs,Ys) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(B),nat,size_size(list(B)),Ys)),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F2,X,aa(B,option(B),some(B),Y)),Xs,Ys) = fun_upd(A,option(B),map_upds(A,B,F2,Xs,Ys),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_6109_lexord__take__index__conv,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R2)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y)))
          & ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),X)),Y) = X ) )
        | ? [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y))))
            & ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I4),X) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I4),Y) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,X),I4)),aa(nat,A,nth(A,Y),I4))),R2)) ) ) ) ).

% lexord_take_index_conv
tff(fact_6110_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R2)))
     => ~ ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ys)))
             => ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I2),Ys) )
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Ys),I2))),R2)) ) ) ) ) ).

% lex_take_index
tff(fact_6111_lexord__lex,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lex(A,R2)))
    <=> ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R2)))
        & ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_6112_lexord__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X4: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R2))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lexord(A,R2))) ) ).

% lexord_irreflexive
tff(fact_6113_lexord__linear,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: list(A),Y: list(A)] :
      ( ! [A4: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B4)),R2))
          | ( A4 = B4 )
          | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4)),R2)) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R2)))
        | ( X = Y )
        | pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),X)),lexord(A,R2))) ) ) ).

% lexord_linear
tff(fact_6114_lexord__Nil__right,axiom,
    ! [A: $tType,X: list(A),R2: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A))),lexord(A,R2))) ).

% lexord_Nil_right
tff(fact_6115_Nil__notin__lex,axiom,
    ! [A: $tType,Ys: list(A),R2: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys)),lex(A,R2))) ).

% Nil_notin_lex
tff(fact_6116_Nil2__notin__lex,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A))),lex(A,R2))) ).

% Nil2_notin_lex
tff(fact_6117_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X4: A,Y2: A,Z3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y2)),R2))
           => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2))
             => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z3)),R2)) ) ) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R2)))
       => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R2)))
         => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs)),lexord(A,R2))) ) ) ) ).

% lexord_partial_trans
tff(fact_6118_List_Olexordp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( lexordp(A,R2,Xs,Ys)
    <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R2))))) ) ).

% List.lexordp_def
tff(fact_6119_lenlex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_acd(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R2))) ).

% lenlex_conv
tff(fact_6120_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ns)),lenlex(A,R2)))
    <=> ( Ns != nil(A) ) ) ).

% Nil_lenlex_iff1
tff(fact_6121_lenlex__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X4: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R2))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lenlex(A,R2))) ) ).

% lenlex_irreflexive
tff(fact_6122_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ns),nil(A))),lenlex(A,R2))) ).

% Nil_lenlex_iff2
tff(fact_6123_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R2)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))) ) ).

% lenlex_length
tff(fact_6124_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_rx(A,fun(A,bool))) = ord_max(A) ) ) ).

% dual_min
tff(fact_6125_listrel1__subset__listrel,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),R4))
     => ( refl_on(A,top_top(set(A)),R4)
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel(A,A,R4))) ) ) ).

% listrel1_subset_listrel
tff(fact_6126_refl__onD,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A] :
      ( refl_on(A,A3,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2)),R2)) ) ) ).

% refl_onD
tff(fact_6127_refl__onD1,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A3,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ) ).

% refl_onD1
tff(fact_6128_refl__onD2,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A3,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3)) ) ) ).

% refl_onD2
tff(fact_6129_ord_Omin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : min(A,Less_eq) = min(A,Less_eq) ).

% ord.min.cong
tff(fact_6130_ord_Omin__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A2: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A2),B2) = A2 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A2),B2) = B2 ) ) ) ).

% ord.min_def
tff(fact_6131_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_6132_refl__on__def_H,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A3,R2)
    <=> ( ! [X3: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),X3),R2))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_ace(set(A),fun(A,fun(A,bool)),A3)),X3)) )
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R2)) ) ) ) ).

% refl_on_def'
tff(fact_6133_refl__on__singleton,axiom,
    ! [A: $tType,X: A] : refl_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_6134_refl__on__domain,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( refl_on(A,A3,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3)) ) ) ) ).

% refl_on_domain
tff(fact_6135_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),I: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys)))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),aa(nat,A,nth(A,Xs),I)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),I)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_6136_map__of__eq__empty__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ! [X3: A] : aa(A,option(B),map_of(A,B,Xys),X3) = none(B)
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% map_of_eq_empty_iff
tff(fact_6137_empty__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ( aTP_Lamp_aat(A,option(B)) = map_of(A,B,Xys) )
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% empty_eq_map_of_iff
tff(fact_6138_map__of__zip__is__None,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X) = none(B) )
      <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% map_of_zip_is_None
tff(fact_6139_dom__map__of__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( dom(A,B,map_of(A,B,zip(A,B,Xs,Ys))) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% dom_map_of_zip
tff(fact_6140_finite__dom__map__of,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B))] : pp(aa(set(A),bool,finite_finite2(A),dom(A,B,map_of(A,B,L)))) ).

% finite_dom_map_of
tff(fact_6141_map__of__Cons__code_I1_J,axiom,
    ! [B: $tType,A: $tType,K: B] : aa(B,option(A),map_of(B,A,nil(product_prod(B,A))),K) = none(A) ).

% map_of_Cons_code(1)
tff(fact_6142_map__of_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType,X2: A] : aa(A,option(B),map_of(A,B,nil(product_prod(A,B))),X2) = none(B) ).

% map_of.simps(1)
tff(fact_6143_finite__graph__map__of,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),graph(A,B,map_of(A,B,Al)))) ).

% finite_graph_map_of
tff(fact_6144_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y) )
     => pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),Y)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs))) ) ).

% map_of_SomeD
tff(fact_6145_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K: A,X: B,L: list(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),X)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L)))
     => ? [X4: B] : aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X4) ) ).

% weak_map_of_SomeI
tff(fact_6146_map__of__zip__inject,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xs) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xs) )
       => ( distinct(B,Xs)
         => ( ( map_of(B,A,zip(B,A,Xs,Ys)) = map_of(B,A,zip(B,A,Xs,Zs)) )
           => ( Ys = Zs ) ) ) ) ) ).

% map_of_zip_inject
tff(fact_6147_finite__range__map__of,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A))] : pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),map_of(B,A,Xys)),top_top(set(B))))) ).

% finite_range_map_of
tff(fact_6148_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
      <=> ? [Y4: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X) = aa(B,option(B),some(B),Y4) ) ) ).

% map_of_zip_is_Some
tff(fact_6149_map__of__eq__None__iff,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),X: B] :
      ( ( aa(B,option(A),map_of(B,A,Xys),X) = none(A) )
    <=> ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(set(product_prod(B,A)),set(B),image2(product_prod(B,A),B,product_fst(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xys)))) ) ).

% map_of_eq_None_iff
tff(fact_6150_map__of__zip__upd,axiom,
    ! [A: $tType,B: $tType,Ys: list(B),Xs: list(A),Zs: list(B),X: A,Y: B,Z: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(A),nat,size_size(list(A)),Xs) )
       => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => ( ( fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X,aa(B,option(B),some(B),Y)) = fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Zs)),X,aa(B,option(B),some(B),Z)) )
           => ( map_of(A,B,zip(A,B,Xs,Ys)) = map_of(A,B,zip(A,B,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_6151_ran__map__of__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( ran(A,B,map_of(A,B,zip(A,B,Xs,Ys))) = aa(list(B),set(B),set2(B),Ys) ) ) ) ).

% ran_map_of_zip
tff(fact_6152_linear__order__on__singleton,axiom,
    ! [A: $tType,X: A] : order_679001287576687338der_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_6153_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xys))
     => ( ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) )
      <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))) ) ) ).

% map_of_eq_Some_iff
tff(fact_6154_map__ident,axiom,
    ! [A: $tType,X2: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ki(A,A)),X2) = X2 ).

% map_ident
tff(fact_6155_list_Omap__disc__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: list(A)] :
      ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),A2) = nil(B) )
    <=> ( A2 = nil(A) ) ) ).

% list.map_disc_iff
tff(fact_6156_Nil__is__map__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( ( nil(A) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) )
    <=> ( Xs = nil(B) ) ) ).

% Nil_is_map_conv
tff(fact_6157_map__is__Nil__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = nil(A) )
    <=> ( Xs = nil(B) ) ) ).

% map_is_Nil_conv
tff(fact_6158_list_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F2: fun(A,B),V2: list(A)] : aa(list(B),list(C),aa(fun(B,C),fun(list(B),list(C)),map(B,C),G),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),V2)) = aa(list(A),list(C),aa(fun(A,C),fun(list(A),list(C)),map(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),F2)),V2) ).

% list.map_comp
tff(fact_6159_List_Omap_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(B,C),G: fun(A,B),List: list(A)] : aa(list(B),list(C),aa(fun(B,C),fun(list(B),list(C)),map(B,C),F2),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),List)) = aa(list(A),list(C),aa(fun(A,C),fun(list(A),list(C)),map(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,F2),G)),List) ).

% List.map.compositionality
tff(fact_6160_map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),Xs: list(C)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),G),Xs)) = aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,F2),G)),Xs) ).

% map_map
tff(fact_6161_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),G: fun(B,A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs) )
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Xs)))
         => ( aa(B,A,F2,X3) = aa(B,A,G,X3) ) ) ) ).

% map_eq_conv
tff(fact_6162_length__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_6163_map__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),N: nat,X: B] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),aa(B,A,F2,X)) ).

% map_replicate
tff(fact_6164_list_Oset__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),V2: list(A)] : aa(list(B),set(B),set2(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),V2)) = aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),V2)) ).

% list.set_map
tff(fact_6165_map__snd__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(A),aa(fun(product_prod(nat,A),A),fun(list(product_prod(nat,A)),list(A)),map(product_prod(nat,A),A),product_snd(nat,A)),enumerate(A,N,Xs)) = Xs ).

% map_snd_enumerate
tff(fact_6166_inj__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_map_eq_map
tff(fact_6167_map__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y: A,Xs: list(A),F2: fun(A,B),V2: B] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),fun_upd(A,B,F2,Y,V2)),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs) ) ) ).

% map_fun_upd
tff(fact_6168_List_Omap_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,C),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),list(C)),comp(list(B),list(C),list(A),aa(fun(B,C),fun(list(B),list(C)),map(B,C),F2)),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G)) = aa(fun(A,C),fun(list(A),list(C)),map(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,F2),G)) ).

% List.map.comp
tff(fact_6169_map__comp__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),comp(list(C),list(B),list(A),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F2)),aa(fun(A,C),fun(list(A),list(C)),map(A,C),G)) = aa(fun(A,B),fun(list(A),list(B)),map(A,B),aa(fun(A,C),fun(A,B),comp(C,B,A,F2),G)) ).

% map_comp_map
tff(fact_6170_size__list__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F2),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs)) = aa(list(B),nat,size_list(B,aa(fun(B,A),fun(B,nat),comp(A,nat,B,F2),G)),Xs) ).

% size_list_map
tff(fact_6171_nth__map,axiom,
    ! [B: $tType,A: $tType,N: nat,Xs: list(A),F2: fun(A,B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs)),N) = aa(A,B,F2,aa(nat,A,nth(A,Xs),N)) ) ) ).

% nth_map
tff(fact_6172_map__fst__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),zip(A,B,Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_6173_map__snd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),zip(A,B,Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_6174_inj__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),top_top(set(list(A))))
    <=> inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_map
tff(fact_6175_inj__mapI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),top_top(set(list(A)))) ) ).

% inj_mapI
tff(fact_6176_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xys))
     => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)))
       => ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_is_SomeI
tff(fact_6177_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,X: A] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xys))
     => ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),X) )
      <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))) ) ) ).

% Some_eq_map_of_iff
tff(fact_6178_nths__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),I6: set(nat)] : nths(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs),I6) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),nths(B,Xs,I6)) ).

% nths_map
tff(fact_6179_map__injective,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
     => ( inj_on(B,A,F2,top_top(set(B)))
       => ( Xs = Ys ) ) ) ).

% map_injective
tff(fact_6180_foldl__map,axiom,
    ! [A: $tType,B: $tType,C: $tType,G: fun(A,fun(B,A)),A2: A,F2: fun(C,B),Xs: list(C)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),G),A2),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F2),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),aa(fun(A,fun(C,A)),fun(A,fun(list(C),A)),foldl(A,C),aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_acf(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G),F2)),A2),Xs) ).

% foldl_map
tff(fact_6181_image__set,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) ).

% image_set
tff(fact_6182_enumerate__Suc__eq,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,aa(nat,nat,suc,N),Xs) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(fun(product_prod(nat,A),product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),map(product_prod(nat,A),product_prod(nat,A)),product_apfst(nat,nat,A,suc)),enumerate(A,N,Xs)) ).

% enumerate_Suc_eq
tff(fact_6183_pair__list__eqI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs) = aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Ys) )
     => ( ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Xs) = aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_6184_list_Omap__ident,axiom,
    ! [A: $tType,T2: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),aTP_Lamp_ki(A,A)),T2) = T2 ).

% list.map_ident
tff(fact_6185_remdups__map__remdups,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : remdups(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),remdups(B,Xs))) = remdups(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) ).

% remdups_map_remdups
tff(fact_6186_list_Osimps_I8_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B)] : aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),nil(A)) = nil(B) ).

% list.simps(8)
tff(fact_6187_rotate1__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate1(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),rotate1(B),Xs)) ).

% rotate1_map
tff(fact_6188_map__update,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),K: nat,Y: B] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Xs),K),Y)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)),K),aa(B,A,F2,Y)) ).

% map_update
tff(fact_6189_map__concat,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(list(B))] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(list(B)),list(B),concat(B),Xs)) = aa(list(list(A)),list(A),concat(A),aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2)),Xs)) ).

% map_concat
tff(fact_6190_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(B,A),Xs: list(B),G: fun(C,A),Ys: list(C)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),G),Ys) )
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_6191_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,X: list(A),Ya: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( ! [Z3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(list(A),set(A),set2(A),Ya)))
           => ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
       => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),X) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ya) ) ) ) ).

% list.map_cong
tff(fact_6192_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,X: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [Z3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(list(A),set(A),set2(A),X)))
         => ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),X) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),X) ) ) ).

% list.map_cong0
tff(fact_6193_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,X: list(A),Xa2: list(A),F2: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z3: A,Za2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(list(A),set(A),set2(A),X)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Za2),aa(list(A),set(A),set2(A),Xa2)))
           => ( ( aa(A,B,F2,Z3) = aa(A,B,Fa,Za2) )
             => ( Z3 = Za2 ) ) ) )
     => ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),X) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),Fa),Xa2) )
       => ( X = Xa2 ) ) ) ).

% list.inj_map_strong
tff(fact_6194_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) )
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xs) ) ) ).

% map_ext
tff(fact_6195_map__idI,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( aa(A,A,F2,X4) = X4 ) )
     => ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),F2),Xs) = Xs ) ) ).

% map_idI
tff(fact_6196_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xs = Ys )
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ys)))
           => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) )
       => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ys) ) ) ) ).

% map_cong
tff(fact_6197_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ys: list(B),F2: fun(A,B)] :
      ( ? [Xs3: list(A)] : Ys = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs3)
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Ys)))
         => ? [Xa3: A] : X3 = aa(A,B,F2,Xa3) ) ) ).

% ex_map_conv
tff(fact_6198_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F2: fun(D,B),Xs: list(D),G: fun(D,C)] : aa(list(product_prod(B,C)),list(A),aa(fun(product_prod(B,C),A),fun(list(product_prod(B,C)),list(A)),map(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H)),zip(B,C,aa(list(D),list(B),aa(fun(D,B),fun(list(D),list(B)),map(D,B),F2),Xs),aa(list(D),list(C),aa(fun(D,C),fun(list(D),list(C)),map(D,C),G),Xs))) = aa(list(D),list(A),aa(fun(D,A),fun(list(D),list(A)),map(D,A),aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_acg(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F2),G)),Xs) ).

% map2_map_map
tff(fact_6199_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),F2),Xs),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),aa(fun(product_prod(C,B),product_prod(A,B)),fun(list(product_prod(C,B)),list(product_prod(A,B))),map(product_prod(C,B),product_prod(A,B)),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_ach(fun(C,A),fun(C,fun(B,product_prod(A,B))),F2))),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_6200_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F2: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F2),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),aa(fun(product_prod(A,C),product_prod(A,B)),fun(list(product_prod(A,C)),list(product_prod(A,B))),map(product_prod(A,C),product_prod(A,B)),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_aci(fun(C,B),fun(A,fun(C,product_prod(A,B))),F2))),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_6201_zip__map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,F2: fun(C,A),Xs: list(C),G: fun(D,B),Ys: list(D)] : zip(A,B,aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),F2),Xs),aa(list(D),list(B),aa(fun(D,B),fun(list(D),list(B)),map(D,B),G),Ys)) = aa(list(product_prod(C,D)),list(product_prod(A,B)),aa(fun(product_prod(C,D),product_prod(A,B)),fun(list(product_prod(C,D)),list(product_prod(A,B))),map(product_prod(C,D),product_prod(A,B)),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_acj(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F2),G))),zip(C,D,Xs,Ys)) ).

% zip_map_map
tff(fact_6202_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F2: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),aa(fun(product_prod(B,C),A),fun(list(product_prod(B,C)),list(A)),map(product_prod(B,C),A),F2),zip(B,C,aa(list(D),list(B),aa(fun(D,B),fun(list(D),list(B)),map(D,B),G),Xs),Ys)) = aa(list(product_prod(D,C)),list(A),aa(fun(product_prod(D,C),A),fun(list(product_prod(D,C)),list(A)),map(product_prod(D,C),A),aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_ack(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F2),G))),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_6203_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),aa(fun(product_prod(B,C),A),fun(list(product_prod(B,C)),list(A)),map(product_prod(B,C),A),F2),zip(B,C,Xs,aa(list(D),list(C),aa(fun(D,C),fun(list(D),list(C)),map(D,C),G),Ys))) = aa(list(product_prod(B,D)),list(A),aa(fun(product_prod(B,D),A),fun(list(product_prod(B,D)),list(A)),map(product_prod(B,D),A),aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_acl(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F2),G))),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_6204_take__map,axiom,
    ! [A: $tType,B: $tType,N: nat,F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Xs)) ).

% take_map
tff(fact_6205_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aTP_Lamp_kv(A,fun(B,A),K)),Lst) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(B),nat,size_size(list(B)),Lst)),K) ).

% map_replicate_const
tff(fact_6206_list_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,nat),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),nat),comp(list(B),nat,list(A),size_list(B,F2)),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G)) = size_list(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F2),G)) ).

% list.size_gen_o_map
tff(fact_6207_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
     => ( inj_on(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)))
       => ( Xs = Ys ) ) ) ).

% map_inj_on
tff(fact_6208_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
     => ( ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_on_map_eq_map
tff(fact_6209_distinct__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( distinct(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
    <=> ( distinct(B,Xs)
        & inj_on(B,A,F2,aa(list(B),set(B),set2(B),Xs)) ) ) ).

% distinct_map
tff(fact_6210_remdups__adj__map__injective,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( remdups_adj(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),remdups_adj(A,Xs)) ) ) ).

% remdups_adj_map_injective
tff(fact_6211_zip__map__fst__snd,axiom,
    ! [B: $tType,A: $tType,Zs: list(product_prod(A,B))] : zip(A,B,aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Zs),aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Zs)) = Zs ).

% zip_map_fst_snd
tff(fact_6212_map__of__eqI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs)) = aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Ys)) )
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs))))
           => ( aa(A,option(B),map_of(A,B,Xs),X4) = aa(A,option(B),map_of(A,B,Ys),X4) ) )
       => ( map_of(A,B,Xs) = map_of(A,B,Ys) ) ) ) ).

% map_of_eqI
tff(fact_6213_map__removeAll__inj,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X: A,Xs: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),removeAll(A,X),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F2,X)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs)) ) ) ).

% map_removeAll_inj
tff(fact_6214_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_6215_inj__mapD,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),top_top(set(list(A))))
     => inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_mapD
tff(fact_6216_zip__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),aa(fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),fun(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C)))),map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_acm(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs)) ).

% zip_assoc
tff(fact_6217_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),aa(fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),fun(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C)))),map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_aco(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs))) ).

% zip_left_commute
tff(fact_6218_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(list(product_prod(B,A)),list(product_prod(A,B))),map(product_prod(B,A),product_prod(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_jn(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_6219_zip__eq__conv,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(product_prod(A,B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( zip(A,B,Xs,Ys) = Zs )
      <=> ( ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Zs) = Xs )
          & ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Zs) = Ys ) ) ) ) ).

% zip_eq_conv
tff(fact_6220_distinct__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] :
          ( distinct(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)))
        <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X)),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))))
            & distinct(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) ) ) ) ).

% distinct_insort_key
tff(fact_6221_map__removeAll__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X: A,Xs: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),removeAll(A,X),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F2,X)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs)) ) ) ).

% map_removeAll_inj_on
tff(fact_6222_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K: A,V1: B,V22: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs))
     => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V1)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
       => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V22)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
tff(fact_6223_inj__on__mapI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(list(A))] :
      ( inj_on(A,B,F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),A3)))
     => inj_on(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),A3) ) ).

% inj_on_mapI
tff(fact_6224_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F2: fun(A,B),X2: A] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs))),X2) = aa(B,option(B),some(B),aa(A,B,F2,X2)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs))),X2) = none(B) ) ) ) ).

% map_of_zip_map
tff(fact_6225_map__fst__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),zip(A,B,Xs,Ys)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))),Xs) ).

% map_fst_zip_take
tff(fact_6226_map__snd__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(B),Ys: list(A)] : aa(list(product_prod(B,A)),list(A),aa(fun(product_prod(B,A),A),fun(list(product_prod(B,A)),list(A)),map(product_prod(B,A),A),product_snd(B,A)),zip(B,A,Xs,Ys)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))),Ys) ).

% map_snd_zip_take
tff(fact_6227_map__of__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,aa(list(product_prod(A,C)),list(product_prod(A,B)),aa(fun(product_prod(A,C),product_prod(A,B)),fun(list(product_prod(A,C)),list(product_prod(A,B))),map(product_prod(A,C),product_prod(A,B)),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_aci(fun(C,B),fun(A,fun(C,product_prod(A,B))),F2))),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F2)),map_of(A,C,Xs)) ).

% map_of_map
tff(fact_6228_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),T2: list(product_prod(A,C)),K: A,X: C] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,T2),K) = aa(C,option(C),some(C),X) )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(A,C)),list(product_prod(B,C)),aa(fun(product_prod(A,C),product_prod(B,C)),fun(list(product_prod(A,C)),list(product_prod(B,C))),map(product_prod(A,C),product_prod(B,C)),aa(fun(A,fun(C,product_prod(B,C))),fun(product_prod(A,C),product_prod(B,C)),product_case_prod(A,C,product_prod(B,C)),aTP_Lamp_acp(fun(A,B),fun(A,fun(C,product_prod(B,C))),F2))),T2)),aa(A,B,F2,K)) = aa(C,option(C),some(C),X) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_6229_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Xs))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_acq(list(product_prod(A,B)),fun(A,fun(B,bool)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_6230_set__relcomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),aa(fun(product_prod(A,C),list(product_prod(A,B))),fun(list(product_prod(A,C)),list(list(product_prod(A,B)))),map(product_prod(A,C),list(product_prod(A,B))),aTP_Lamp_acs(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_6231_total__on__singleton,axiom,
    ! [A: $tType,X: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_6232_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y222: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_6233_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X21),aa(list(A),set(A),set2(A),X222)) ).

% list.simps(15)
tff(fact_6234_nth__Cons__Suc,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).

% nth_Cons_Suc
tff(fact_6235_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_6236_take__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) ).

% take_Suc_Cons
tff(fact_6237_nths__singleton,axiom,
    ! [A: $tType,A3: set(nat),X: A] :
      ( ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) )
      & ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A3) = nil(A) ) ) ) ).

% nths_singleton
tff(fact_6238_Cons__listrel1__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),listrel1(A,R2)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2))) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_6239_lexord__cons__cons,axiom,
    ! [A: $tType,A2: A,X: list(A),B2: A,Y: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y))),lexord(A,R2)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
        | ( ( A2 = B2 )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R2))) ) ) ) ).

% lexord_cons_cons
tff(fact_6240_lexord__Nil__left,axiom,
    ! [A: $tType,Y: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Y)),lexord(A,R2)))
    <=> ? [A7: A,X3: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),X3) ) ).

% lexord_Nil_left
tff(fact_6241_enumerate__simps_I2_J,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : enumerate(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(product_prod(nat,B)),list(product_prod(nat,B)),aa(product_prod(nat,B),fun(list(product_prod(nat,B)),list(product_prod(nat,B))),cons(product_prod(nat,B)),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),N),X)),enumerate(B,aa(nat,nat,suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_6242_map__upds__Cons,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A2: A,As: list(A),B2: B,Bs: list(B)] : map_upds(A,B,M,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),As),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs)) = map_upds(A,B,fun_upd(A,option(B),M,A2,aa(B,option(B),some(B),B2)),As,Bs) ).

% map_upds_Cons
tff(fact_6243_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),zip(A,B,Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_6244_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xs: list(A),V2: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_6245_take__Cons__numeral,axiom,
    ! [A: $tType,V2: num,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(num,nat,numeral_numeral(nat),V2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat))),Xs)) ).

% take_Cons_numeral
tff(fact_6246_Cons__in__lex,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),lex(A,R2)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R2))) ) ) ) ).

% Cons_in_lex
tff(fact_6247_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(list(A)),list(A),concat(A),aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),aTP_Lamp_act(fun(B,A),fun(B,list(A)),F2)),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) ).

% concat_map_singleton
tff(fact_6248_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_6249_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),aa(fun(A,product_prod(A,A)),fun(list(A),list(product_prod(A,A))),map(A,product_prod(A,A)),aTP_Lamp_jl(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_6250_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(A),list(list(product_prod(A,B))),aa(fun(A,list(product_prod(A,B))),fun(list(A),list(list(product_prod(A,B)))),map(A,list(product_prod(A,B))),aTP_Lamp_acu(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_6251_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : aa(list(list(A)),list(list(A)),product_lists(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),Xss)) = aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(A),list(list(list(A))),aa(fun(A,list(list(A))),fun(list(A),list(list(list(A)))),map(A,list(list(A))),aTP_Lamp_acv(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_6252_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aTP_Lamp_acx(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_6253_map__eq__Cons__conv,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ? [Z2: B,Zs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z2),Zs3) )
          & ( aa(B,A,F2,Z2) = Y )
          & ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs3) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_6254_Cons__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),F2: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
    <=> ? [Z2: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z2),Zs3) )
          & ( X = aa(B,A,F2,Z2) )
          & ( Xs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs3) ) ) ) ).

% Cons_eq_map_conv
tff(fact_6255_map__eq__Cons__D,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
     => ? [Z3: B,Zs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z3),Zs2) )
          & ( aa(B,A,F2,Z3) = Y )
          & ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_6256_Cons__eq__map__D,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),F2: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys) )
     => ? [Z3: B,Zs2: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z3),Zs2) )
          & ( X = aa(B,A,F2,Z3) )
          & ( Xs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_6257_list_Osimps_I9_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X21: A,X222: list(A)] : aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,F2,X21)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),X222)) ).

% list.simps(9)
tff(fact_6258_total__lexord,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,top_top(set(A)),R2)
     => total_on(list(A),top_top(set(list(A))),lexord(A,R2)) ) ).

% total_lexord
tff(fact_6259_listrelp_OCons,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X),Y))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),Xs),Ys))
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))) ) ) ).

% listrelp.Cons
tff(fact_6260_list__update__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),zero_zero(nat)),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs) ).

% list_update_code(2)
tff(fact_6261_replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,suc,N)),X) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) ).

% replicate_Suc
tff(fact_6262_insort__key_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Y: B,Ys: list(B)] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y)))
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X)),aa(B,A,F2,Y)))
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Ys)) ) ) ) ) ).

% insort_key.simps(2)
tff(fact_6263_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I: nat,V2: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),I),V2) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_acy(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xs),V2),I) ).

% list_update.simps(2)
tff(fact_6264_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F3: fun(A,B),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs2))
     => ~ ! [F3: fun(A,B),A4: A,As2: list(A),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As2)),Bs2)) ) ).

% map_tailrec_rev.cases
tff(fact_6265_shuffles_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3)
     => ( ! [Xs2: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A))
       => ~ ! [X4: A,Xs2: list(A),Y2: A,Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)) ) ) ).

% shuffles.cases
tff(fact_6266_splice_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3)
     => ~ ! [X4: A,Xs2: list(A),Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Ys3) ) ).

% splice.cases
tff(fact_6267_distinct__singleton,axiom,
    ! [A: $tType,X: A] : distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% distinct_singleton
tff(fact_6268_set__ConsD,axiom,
    ! [A: $tType,Y: A,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))))
     => ( ( Y = X )
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% set_ConsD
tff(fact_6269_list_Oset__cases,axiom,
    ! [A: $tType,E3: A,A2: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),E3),aa(list(A),set(A),set2(A),A2)))
     => ( ! [Z22: list(A)] : A2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Z22)
       => ~ ! [Z1: A,Z22: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z1),Z22) )
             => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),E3),aa(list(A),set(A),set2(A),Z22))) ) ) ) ).

% list.set_cases
tff(fact_6270_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X21),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)))) ).

% list.set_intros(1)
tff(fact_6271_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),X222)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)))) ) ).

% list.set_intros(2)
tff(fact_6272_Cons__in__shuffles__rightI,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z: A] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Zs)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Ys)))) ) ).

% Cons_in_shuffles_rightI
tff(fact_6273_Cons__in__shuffles__leftI,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z: A] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Zs)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Xs),Ys))) ) ).

% Cons_in_shuffles_leftI
tff(fact_6274_not__Cons__self2,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) != Xs ).

% not_Cons_self2
tff(fact_6275_remove1_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( remove1(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = Xs ) )
      & ( ( X != Y )
       => ( remove1(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),remove1(A,X,Xs)) ) ) ) ).

% remove1.simps(2)
tff(fact_6276_distinct__length__2__or__more,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Xs)))
    <=> ( ( A2 != B2 )
        & distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Xs))
        & distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Xs)) ) ) ).

% distinct_length_2_or_more
tff(fact_6277_removeAll_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( aa(list(A),list(A),removeAll(A,X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = aa(list(A),list(A),removeAll(A,X),Xs) ) )
      & ( ( X != Y )
       => ( aa(list(A),list(A),removeAll(A,X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),removeAll(A,X),Xs)) ) ) ) ).

% removeAll.simps(2)
tff(fact_6278_list__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),bool)] :
      ( ( Xs != nil(A) )
     => ( ! [X4: A] : pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))))
       => ( ! [X4: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( pp(aa(list(A),bool,P,Xs2))
               => pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2))) ) )
         => pp(aa(list(A),bool,P,Xs)) ) ) ) ).

% list_nonempty_induct
tff(fact_6279_list__induct2_H,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
     => ( ! [X4: A,Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),nil(B)))
       => ( ! [Y2: B,Ys3: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3)))
         => ( ! [X4: A,Xs2: list(A),Y2: B,Ys3: list(B)] :
                ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3))) )
           => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_6280_neq__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
    <=> ? [Y4: A,Ys4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) ).

% neq_Nil_conv
tff(fact_6281_remdups__adj_Ocases,axiom,
    ! [A: $tType,X: list(A)] :
      ( ( X != nil(A) )
     => ( ! [X4: A] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))
       => ~ ! [X4: A,Y2: A,Xs2: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_6282_transpose_Ocases,axiom,
    ! [A: $tType,X: list(list(A))] :
      ( ( X != nil(list(A)) )
     => ( ! [Xss2: list(list(A))] : X != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)
       => ~ ! [X4: A,Xs2: list(A),Xss2: list(list(A))] : X != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Xss2) ) ) ).

% transpose.cases
tff(fact_6283_min__list_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A)] :
          ( ! [X4: A,Xs2: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)
         => ( X = nil(A) ) ) ) ).

% min_list.cases
tff(fact_6284_list_Oexhaust,axiom,
    ! [A: $tType,Y: list(A)] :
      ( ( Y != nil(A) )
     => ~ ! [X212: A,X223: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X212),X223) ) ).

% list.exhaust
tff(fact_6285_list_OdiscI,axiom,
    ! [A: $tType,List: list(A),X21: A,X222: list(A)] :
      ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) )
     => ( List != nil(A) ) ) ).

% list.discI
tff(fact_6286_list_Odistinct_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) ).

% list.distinct(1)
tff(fact_6287_remdups__adj_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ) )
      & ( ( X != Y )
       => ( remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) ) ) ) ).

% remdups_adj.simps(3)
tff(fact_6288_length__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_6289_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y4: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_6290_Suc__length__conv,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,N) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y4: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_6291_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
       => ( ! [X4: A,Xs2: list(A),Y2: B,Ys3: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3))) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_6292_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),bool)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,nil(A)),nil(B)),nil(C)))
         => ( ! [X4: A,Xs2: list(A),Y2: B,Ys3: list(B),Z3: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,Xs2),Ys3),Zs2))
                   => pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2))) ) ) )
           => pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_6293_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws2: list(D),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D),bool))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs) = aa(list(D),nat,size_size(list(D)),Ws2) )
         => ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,nil(A)),nil(B)),nil(C)),nil(D)))
           => ( ! [X4: A,Xs2: list(A),Y2: B,Ys3: list(B),Z3: C,Zs2: list(C),W: D,Ws: list(D)] :
                  ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
                 => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
                   => ( ( aa(list(C),nat,size_size(list(C)),Zs2) = aa(list(D),nat,size_size(list(D)),Ws) )
                     => ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,Xs2),Ys3),Zs2),Ws))
                       => pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),W),Ws))) ) ) ) )
             => pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,Xs),Ys),Zs),Ws2)) ) ) ) ) ) ).

% list_induct4
tff(fact_6294_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)))
     => ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys) ) ) ).

% impossible_Cons
tff(fact_6295_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))) ).

% set_subset_Cons
tff(fact_6296_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_6297_total__on__def,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A3,R2)
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A3))
             => ( ( X3 != Xa3 )
               => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),R2))
                  | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3)),R2)) ) ) ) ) ) ).

% total_on_def
tff(fact_6298_total__onI,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
           => ( ( X4 != Y2 )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y2)),R2))
                | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X4)),R2)) ) ) ) )
     => total_on(A,A3,R2) ) ).

% total_onI
tff(fact_6299_successively_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
      ( ! [P6: fun(A,fun(A,bool))] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P6),nil(A))
     => ( ! [P6: fun(A,fun(A,bool)),X4: A] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)))
       => ~ ! [P6: fun(A,fun(A,bool)),X4: A,Y2: A,Xs2: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ).

% successively.cases
tff(fact_6300_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: product_prod(fun(A,B),list(A))] :
          ( ! [F3: fun(A,B),X4: A] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)))
         => ( ! [F3: fun(A,B),X4: A,Y2: A,Zs2: list(A)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)))
           => ~ ! [A4: fun(A,B)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A4),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_6301_sorted__wrt_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
      ( ! [P6: fun(A,fun(A,bool))] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P6),nil(A))
     => ~ ! [P6: fun(A,fun(A,bool)),X4: A,Ys3: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3)) ) ).

% sorted_wrt.cases
tff(fact_6302_zip__eq__ConsE,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
      ( ( zip(A,B,Xs,Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),Xy),Xys) )
     => ~ ! [X4: A,Xs4: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
           => ! [Y2: B,Ys5: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys5) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2) )
                 => ( Xys != zip(A,B,Xs4,Ys5) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6303_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
        & distinct(A,Xs) ) ) ).

% distinct.simps(2)
tff(fact_6304_list__update__code_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I: nat,Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(nat,nat,suc,I)),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),Y)) ).

% list_update_code(3)
tff(fact_6305_remdups__adj_Osimps_I2_J,axiom,
    ! [A: $tType,X: A] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).

% remdups_adj.simps(2)
tff(fact_6306_remdups__adj_Oelims,axiom,
    ! [A: $tType,X: list(A),Y: list(A)] :
      ( ( remdups_adj(A,X) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != nil(A) ) )
       => ( ! [X4: A] :
              ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) )
             => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) ) )
         => ~ ! [X4: A,Y2: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
               => ~ ( ( ( X4 = Y2 )
                     => ( Y = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)) ) )
                    & ( ( X4 != Y2 )
                     => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ) ) ) ) ) ).

% remdups_adj.elims
tff(fact_6307_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Y: A,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,Xs,Ys))),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))) ).

% Cons_shuffles_subset2
tff(fact_6308_Cons__shuffles__subset1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,Ys))),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys))) ).

% Cons_shuffles_subset1
tff(fact_6309_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A)] : shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys))) ).

% shuffles.simps(3)
tff(fact_6310_foldl__Cons,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,B)),A2: B,X: A,Xs: list(A)] : aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F2),A2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),B,aa(B,fun(list(A),B),aa(fun(B,fun(A,B)),fun(B,fun(list(A),B)),foldl(B,A),F2),aa(A,B,aa(B,fun(A,B),F2,A2),X)),Xs) ).

% foldl_Cons
tff(fact_6311_insort__key_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),nil(B)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)) ) ).

% insort_key.simps(1)
tff(fact_6312_shufflesE,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => ( ( ( Zs = Xs )
         => ( Ys != nil(A) ) )
       => ( ( ( Zs = Ys )
           => ( Xs != nil(A) ) )
         => ( ! [X4: A,Xs4: list(A)] :
                ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
               => ! [Z3: A,Zs4: list(A)] :
                    ( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z3),Zs4) )
                   => ( ( X4 = Z3 )
                     => ~ pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs4),shuffles(A,Xs4,Ys))) ) ) )
           => ~ ! [Y2: A,Ys5: list(A)] :
                  ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
                 => ! [Z3: A,Zs4: list(A)] :
                      ( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z3),Zs4) )
                     => ( ( Y2 = Z3 )
                       => ~ pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs4),shuffles(A,Xs,Ys5))) ) ) ) ) ) ) ) ).

% shufflesE
tff(fact_6313_inj__on__Cons1,axiom,
    ! [A: $tType,X: A,A3: set(list(A))] : inj_on(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),A3) ).

% inj_on_Cons1
tff(fact_6314_inj__split__Cons,axiom,
    ! [A: $tType,X6: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_acw(list(A),fun(A,list(A)))),X6) ).

% inj_split_Cons
tff(fact_6315_remdups_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( remdups(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = remdups(A,Xs) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( remdups(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),remdups(A,Xs)) ) ) ) ).

% remdups.simps(2)
tff(fact_6316_total__lenlex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,top_top(set(A)),R2)
     => total_on(list(A),top_top(set(list(A))),lenlex(A,R2)) ) ).

% total_lenlex
tff(fact_6317_listrel1I2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),X: A] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),listrel1(A,R2))) ) ).

% listrel1I2
tff(fact_6318_find_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( find(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,option(A),some(A),X) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( find(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = find(A,P,Xs) ) ) ) ).

% find.simps(2)
tff(fact_6319_List_Obind__def,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,list(B))] : bind(A,B,Xs,F2) = aa(list(list(B)),list(B),concat(B),aa(list(A),list(list(B)),aa(fun(A,list(B)),fun(list(A),list(list(B))),map(A,list(B)),F2),Xs)) ).

% List.bind_def
tff(fact_6320_listrelp_Osimps,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A1: list(A),A22: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),A1),A22))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X3: A,Y4: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys4) )
            & pp(aa(B,bool,aa(A,fun(B,bool),R2,X3),Y4))
            & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),Xs3),Ys4)) ) ) ) ).

% listrelp.simps
tff(fact_6321_listrelp_Ocases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A1: list(A),A22: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),A1),A22))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X4: A,Y2: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3) )
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X4),Y2))
                   => ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R2),Xs2),Ys3)) ) ) ) ) ) ).

% listrelp.cases
tff(fact_6322_arg__min__list_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),X: A] : arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = X ) ).

% arg_min_list.simps(1)
tff(fact_6323_take__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),nil(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_acz(A,fun(list(A),fun(nat,list(A))),X),Xs),N) ).

% take_Cons
tff(fact_6324_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ys: list(A),Xs: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs))))
     => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(A),list(list(A)),subseqs(A),Xs)))) ) ).

% Cons_in_subseqsD
tff(fact_6325_nth__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = case_nat(A,X,nth(A,Xs),N) ).

% nth_Cons
tff(fact_6326_ord_Olexordp_Omono,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool))] : pp(aa(fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),bool,order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),aTP_Lamp_ada(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Less))) ).

% ord.lexordp.mono
tff(fact_6327_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),X: A,Y: A,Zs: list(A)] : arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))),X,arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) ) ).

% arg_min_list.simps(2)
tff(fact_6328_Suc__le__length__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ? [X3: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_6329_distinct__set__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(set(A),aa(list(list(A)),list(set(A)),aa(fun(list(A),set(A)),fun(list(list(A)),list(set(A))),map(list(A),set(A)),set2(A)),aa(list(A),list(list(A)),subseqs(A),Xs))) ) ).

% distinct_set_subseqs
tff(fact_6330_insort__is__Cons,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Xs: list(B),F2: fun(B,A),A2: B] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(list(B),set(B),set2(B),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,A2)),aa(B,A,F2,X4))) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),A2),Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_6331_map__of__Cons__code_I2_J,axiom,
    ! [C: $tType,B: $tType,L: B,K: B,V2: C,Ps: list(product_prod(B,C))] :
      ( ( ( L = K )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V2)),Ps)),K) = aa(C,option(C),some(C),V2) ) )
      & ( ( L != K )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V2)),Ps)),K) = aa(B,option(C),map_of(B,C,Ps),K) ) ) ) ).

% map_of_Cons_code(2)
tff(fact_6332_listrel1I1,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))),listrel1(A,R2))) ) ).

% listrel1I1
tff(fact_6333_Cons__listrel1E1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys)),listrel1(A,R2)))
     => ( ! [Y2: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs) )
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R2)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2)),listrel1(A,R2))) ) ) ) ).

% Cons_listrel1E1
tff(fact_6334_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),listrel1(A,R2)))
     => ( ! [X4: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys) )
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y)),R2)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs2),Ys)),listrel1(A,R2))) ) ) ) ).

% Cons_listrel1E2
tff(fact_6335_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),listrel(A,B,R2)))
     => ~ ! [X4: A,Xs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
           => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y)),R2))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys)),listrel(A,B,R2))) ) ) ) ).

% listrel_Cons2
tff(fact_6336_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),Xs)),listrel(A,B,R2)))
     => ~ ! [Y2: B,Ys3: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3) )
           => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y2)),R2))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Ys),Ys3)),listrel(A,B,R2))) ) ) ) ).

% listrel_Cons1
tff(fact_6337_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,R2: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R2))
     => ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R2)))
       => pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),listrel(A,B,R2))) ) ) ).

% listrel.Cons
tff(fact_6338_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y)),one_one(nat)) ) )
      & ( ( X != Y )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y) = aa(A,nat,count_list(A,Xs),Y) ) ) ) ).

% count_list.simps(2)
tff(fact_6339_the__elem__set,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = X ).

% the_elem_set
tff(fact_6340_rtrancl__listrel1__ConsI1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),X: A] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),transitive_rtrancl(list(A),listrel1(A,R2)))) ) ).

% rtrancl_listrel1_ConsI1
tff(fact_6341_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => pp(aa(fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),bool,order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),aTP_Lamp_adb(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))))) ) ).

% lexordp.mono
tff(fact_6342_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,N: nat,X: A,Ys: list(B)] : zip(A,B,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X),Ys) = aa(list(B),list(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(list(B),list(product_prod(A,B))),map(B,product_prod(A,B)),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Ys)) ).

% zip_replicate1
tff(fact_6343_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_6344_nth__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% nth_Cons'
tff(fact_6345_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),N: nat,Y: B] : zip(A,B,Xs,aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),Y)) = aa(list(A),list(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_jn(B,fun(A,product_prod(A,B))),Y)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) ).

% zip_replicate2
tff(fact_6346_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22)),listrel(A,B,R2)))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X4: A,Y2: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3) )
                 => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2)),R2))
                   => ~ pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys3)),listrel(A,B,R2))) ) ) ) ) ) ).

% listrel.cases
tff(fact_6347_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22)),listrel(A,B,R2)))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X3: A,Y4: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys4) )
            & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4)),R2))
            & pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys4)),listrel(A,B,R2))) ) ) ) ).

% listrel.simps
tff(fact_6348_remdups__adj__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( remdups_adj(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( remdups_adj(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ) ).

% remdups_adj_replicate
tff(fact_6349_remdups__adj__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( remdups_adj(A,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
     => ( Xs = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),Xs)),X) ) ) ).

% remdups_adj_singleton
tff(fact_6350_Id__on__set,axiom,
    ! [A: $tType,Xs: list(A)] : id_on(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),aa(fun(A,product_prod(A,A)),fun(list(A),list(product_prod(A,A))),map(A,product_prod(A,A)),aTP_Lamp_jl(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_6351_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : aa(list(A),nat,size_list(A,X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),aa(list(A),nat,size_list(A,X),X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6352_shuffles_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa2) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xa2),bot_bot(set(list(A)))) ) )
       => ( ( ( Xa2 = nil(A) )
           => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),X),bot_bot(set(list(A)))) ) )
         => ~ ! [X4: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
               => ! [Y2: A,Ys3: list(A)] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
                   => ( Y != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2),Ys3))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_6353_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6354_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A),N: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6355_take__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),Xs)) ) ) ) ).

% take_Cons'
tff(fact_6356_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),Y) )
    <=> ( ( X = Y )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
        & ( Xs = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_6357_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),M: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,M) )
     => ( map_of(A,B,aa(list(A),list(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aTP_Lamp_abt(fun(A,option(B)),fun(A,product_prod(A,B)),M)),Xs)) = M ) ) ).

% map_of_map_keys
tff(fact_6358_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),N: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns))),lenlex(A,R2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),N)),R2)) )
        | ( ( M = N )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R2))) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6359_map__of_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),Ps: list(product_prod(A,B))] : map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),P2),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P2),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P2))) ).

% map_of.simps(2)
tff(fact_6360_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2)))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),transitive_rtrancl(list(A),listrel1(A,R2)))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_6361_Pow__set_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : pow(B,aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs))) = aa(set(set(B)),set(set(B)),aa(set(set(B)),fun(set(set(B)),set(set(B))),sup_sup(set(set(B))),pow(B,aa(list(B),set(B),set2(B),Xs))),aa(set(set(B)),set(set(B)),image2(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X)),pow(B,aa(list(B),set(B),set2(B),Xs)))) ).

% Pow_set(2)
tff(fact_6362_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_6363_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(list(A),list(product_prod(A,B))),map(A,product_prod(A,B)),aTP_Lamp_xc(fun(A,B),fun(A,product_prod(A,B)),F2)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F2),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_6364_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(A),list(list(product_prod(A,B))),aa(fun(A,list(product_prod(A,B))),fun(list(A),list(list(product_prod(A,B)))),map(A,list(product_prod(A,B))),aTP_Lamp_acu(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_6365_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A3: set(A)] : set_Cons(A,A3,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))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_adc(A,list(A))),A3) ).

% set_Cons_sing_Nil
tff(fact_6366_n__lists__Nil,axiom,
    ! [A: $tType,N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) )
      & ( ( N != zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).

% n_lists_Nil
tff(fact_6367_listset_Osimps_I2_J,axiom,
    ! [A: $tType,A3: set(A),As3: list(set(A))] : listset(A,aa(list(set(A)),list(set(A)),aa(set(A),fun(list(set(A)),list(set(A))),cons(set(A)),A3),As3)) = set_Cons(A,A3,listset(A,As3)) ).

% listset.simps(2)
tff(fact_6368_subseqs_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(list(A)),subseqs(A),nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% subseqs.simps(1)
tff(fact_6369_product__lists_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(list(A)),list(list(A)),product_lists(A),nil(list(A))) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% product_lists.simps(1)
tff(fact_6370_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_6371_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I)),J))
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_6372_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),J))
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_6373_set__Cons__def,axiom,
    ! [A: $tType,A3: set(A),XS: set(list(A))] : set_Cons(A,A3,XS) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(set(list(A)),fun(list(A),bool),aTP_Lamp_add(set(A),fun(set(list(A)),fun(list(A),bool)),A3),XS)) ).

% set_Cons_def
tff(fact_6374_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% concat_inth
tff(fact_6375_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
     => ( map_upds(A,B,M,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),Ys) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys),X,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_6376_append_Oassoc,axiom,
    ! [A: $tType,A2: list(A),B2: list(A),C2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),B2)),C2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),B2),C2)) ).

% append.assoc
tff(fact_6377_append__assoc,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Zs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs)) ).

% append_assoc
tff(fact_6378_append__same__eq,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Xs) )
    <=> ( Ys = Zs ) ) ).

% append_same_eq
tff(fact_6379_same__append__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs) )
    <=> ( Ys = Zs ) ) ).

% same_append_eq
tff(fact_6380_append__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = nil(A) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% append_is_Nil_conv
tff(fact_6381_Nil__is__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_is_append_conv
tff(fact_6382_self__append__conv2,axiom,
    ! [A: $tType,Y: list(A),Xs: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Y) )
    <=> ( Xs = nil(A) ) ) ).

% self_append_conv2
tff(fact_6383_append__self__conv2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Ys )
    <=> ( Xs = nil(A) ) ) ).

% append_self_conv2
tff(fact_6384_self__append__conv,axiom,
    ! [A: $tType,Y: list(A),Ys: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Y),Ys) )
    <=> ( Ys = nil(A) ) ) ).

% self_append_conv
tff(fact_6385_append__self__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Xs )
    <=> ( Ys = nil(A) ) ) ).

% append_self_conv
tff(fact_6386_append__Nil2,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),nil(A)) = Xs ).

% append_Nil2
tff(fact_6387_append_Oright__neutral,axiom,
    ! [A: $tType,A2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),nil(A)) = A2 ).

% append.right_neutral
tff(fact_6388_append__eq__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Us: list(A),Vs: list(A)] :
      ( ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        | ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_6389_map__append,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys)) ).

% map_append
tff(fact_6390_concat__append,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] : aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),Xs)),aa(list(list(A)),list(A),concat(A),Ys)) ).

% concat_append
tff(fact_6391_removeAll__append,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),removeAll(A,X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),removeAll(A,X),Xs)),aa(list(A),list(A),removeAll(A,X),Ys)) ).

% removeAll_append
tff(fact_6392_foldl__append,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,A)),A2: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A2),Xs)),Ys) ).

% foldl_append
tff(fact_6393_append1__eq__conv,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))) )
    <=> ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
tff(fact_6394_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_append
tff(fact_6395_set__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_append
tff(fact_6396_zip__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Us: list(B),Ys: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Us) )
     => ( zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us),Vs)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),zip(A,B,Xs,Us)),zip(A,B,Ys,Vs)) ) ) ).

% zip_append
tff(fact_6397_fun__upds__append2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),Zs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M,Xs,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = map_upds(A,B,M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_6398_fun__upds__append__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs),Ys) = map_upds(A,B,M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_6399_size__list__append,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_list(A,F2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_list(A,F2),Xs)),aa(list(A),nat,size_list(A,F2),Ys)) ).

% size_list_append
tff(fact_6400_bind__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: B,Xs: list(B),F2: fun(B,list(A))] : bind(B,A,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),F2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F2,X)),bind(B,A,Xs,F2)) ).

% bind_simps(2)
tff(fact_6401_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% nth_append_length
tff(fact_6402_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys),N) ).

% nth_append_length_plus
tff(fact_6403_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A] : aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),aa(list(A),nat,size_size(list(A)),Xs)),Y) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ).

% list_update_length
tff(fact_6404_take__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),Ys)) ).

% take_append
tff(fact_6405_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_lessThan(nat,aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_lessThan(nat,K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_6406_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_atMost(nat,aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_atMost(nat,K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_6407_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( distinct(A,Xs)
        & distinct(A,Ys)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_6408_map__eq__append__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ? [Us2: list(B),Vs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs2) )
          & ( Ys = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Us2) )
          & ( Zs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Vs2) ) ) ) ).

% map_eq_append_conv
tff(fact_6409_append__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Ys: list(A),Zs: list(A),F2: fun(B,A),Xs: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) )
    <=> ? [Us2: list(B),Vs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs2) )
          & ( Ys = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Us2) )
          & ( Zs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Vs2) ) ) ) ).

% append_eq_map_conv
tff(fact_6410_replicate__app__Cons__same,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),Xs)) ).

% replicate_app_Cons_same
tff(fact_6411_split__list,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ? [Ys3: list(A),Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) ) ).

% split_list
tff(fact_6412_split__list__last,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ? [Ys3: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Zs2))) ) ) ).

% split_list_last
tff(fact_6413_split__list__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ? [Ys3: list(A),X4: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs2))
          & pp(aa(A,bool,P,X4)) ) ) ).

% split_list_prop
tff(fact_6414_split__list__first,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ? [Ys3: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Ys3))) ) ) ).

% split_list_first
tff(fact_6415_split__list__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ~ ! [Ys3: list(A),X4: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs2))
           => ~ pp(aa(A,bool,P,X4)) ) ) ).

% split_list_propE
tff(fact_6416_append__Cons__eq__iff,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A),Xs5: list(A),Ys6: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Ys)))
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys6)) )
        <=> ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_6417_in__set__conv__decomp,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [Ys4: list(A),Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) ) ).

% in_set_conv_decomp
tff(fact_6418_split__list__last__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ? [Ys3: list(A),X4: A,Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs2)) )
          & pp(aa(A,bool,P,X4))
          & ! [Xa: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),aa(list(A),set(A),set2(A),Zs2)))
             => ~ pp(aa(A,bool,P,Xa)) ) ) ) ).

% split_list_last_prop
tff(fact_6419_split__list__first__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ? [Ys3: list(A),X4: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs2))
          & pp(aa(A,bool,P,X4))
          & ! [Xa: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),aa(list(A),set(A),set2(A),Ys3)))
             => ~ pp(aa(A,bool,P,Xa)) ) ) ) ).

% split_list_first_prop
tff(fact_6420_split__list__last__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ~ ! [Ys3: list(A),X4: A,Zs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs2)) )
           => ( pp(aa(A,bool,P,X4))
             => ~ ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),aa(list(A),set(A),set2(A),Zs2)))
                   => ~ pp(aa(A,bool,P,Xa)) ) ) ) ) ).

% split_list_last_propE
tff(fact_6421_split__list__first__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ~ ! [Ys3: list(A),X4: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs2))
           => ( pp(aa(A,bool,P,X4))
             => ~ ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),aa(list(A),set(A),set2(A),Ys3)))
                   => ~ pp(aa(A,bool,P,Xa)) ) ) ) ) ).

% split_list_first_propE
tff(fact_6422_in__set__conv__decomp__last,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Zs3))) ) ) ).

% in_set_conv_decomp_last
tff(fact_6423_in__set__conv__decomp__first,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Ys4))) ) ) ).

% in_set_conv_decomp_first
tff(fact_6424_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X3)) )
    <=> ? [Ys4: list(A),X3: A,Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs3)) )
          & pp(aa(A,bool,P,X3))
          & ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(list(A),set(A),set2(A),Zs3)))
             => ~ pp(aa(A,bool,P,Xa3)) ) ) ) ).

% split_list_last_prop_iff
tff(fact_6425_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X3)) )
    <=> ? [Ys4: list(A),X3: A] :
          ( ? [Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs3))
          & pp(aa(A,bool,P,X3))
          & ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(list(A),set(A),set2(A),Ys4)))
             => ~ pp(aa(A,bool,P,Xa3)) ) ) ) ).

% split_list_first_prop_iff
tff(fact_6426_concat_Osimps_I2_J,axiom,
    ! [A: $tType,X: list(A),Xs: list(list(A))] : aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),aa(list(list(A)),list(A),concat(A),Xs)) ).

% concat.simps(2)
tff(fact_6427_append__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% append_Cons
tff(fact_6428_Cons__eq__appendI,axiom,
    ! [A: $tType,X: A,Xs1: list(A),Ys: list(A),Xs: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) = Ys )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Zs) )
       => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) ) ) ) ).

% Cons_eq_appendI
tff(fact_6429_rev__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),bool)] :
      ( ( Xs != nil(A) )
     => ( ! [X4: A] : pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))))
       => ( ! [X4: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( pp(aa(list(A),bool,P,Xs2))
               => pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))))) ) )
         => pp(aa(list(A),bool,P,Xs)) ) ) ) ).

% rev_nonempty_induct
tff(fact_6430_append__eq__Cons__conv,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
    <=> ( ( ( Ys = nil(A) )
          & ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
        | ? [Ys7: list(A)] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys7) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys7),Zs) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_6431_Cons__eq__append__conv,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ( ( ( Ys = nil(A) )
          & ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = Zs ) )
        | ? [Ys7: list(A)] :
            ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys7) = Ys )
            & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys7),Zs) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_6432_rev__exhaust,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ~ ! [Ys3: list(A),Y2: A] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))) ) ).

% rev_exhaust
tff(fact_6433_rev__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
      ( pp(aa(list(A),bool,P,nil(A)))
     => ( ! [X4: A,Xs2: list(A)] :
            ( pp(aa(list(A),bool,P,Xs2))
           => pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))))) )
       => pp(aa(list(A),bool,P,Xs)) ) ) ).

% rev_induct
tff(fact_6434_concat__eq__appendD,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(list(A)),list(A),concat(A),Xss) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
     => ( ( Xss != nil(list(A)) )
       => ? [Xss1: list(list(A)),Xs2: list(A),Xs4: list(A),Xss22: list(list(A))] :
            ( ( Xss = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),Xs4)),Xss22)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),Xss1)),Xs2) )
            & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),aa(list(list(A)),list(A),concat(A),Xss22)) ) ) ) ) ).

% concat_eq_appendD
tff(fact_6435_concat__eq__append__conv,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(list(A)),list(A),concat(A),Xss) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ( ( ( Xss = nil(list(A)) )
         => ( ( Ys = nil(A) )
            & ( Zs = nil(A) ) ) )
        & ( ( Xss != nil(list(A)) )
         => ? [Xss12: list(list(A)),Xs3: list(A),Xs6: list(A),Xss23: list(list(A))] :
              ( ( Xss = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),Xs6)),Xss23)) )
              & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),Xss12)),Xs3) )
              & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs6),aa(list(list(A)),list(A),concat(A),Xss23)) ) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_6436_lex__append__leftI,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),R2: set(product_prod(A,A)),Xs: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R2)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R2))) ) ).

% lex_append_leftI
tff(fact_6437_append__listrel1I,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Us: list(A),Vs: list(A)] :
      ( ( ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2)))
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Vs)),listrel1(A,R2))) ) )
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs))),listrel1(A,R2))) ) ).

% append_listrel1I
tff(fact_6438_lexord__append__leftI,axiom,
    ! [A: $tType,U: list(A),V2: list(A),R2: set(product_prod(A,A)),X: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V2)),lexord(A,R2)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V2))),lexord(A,R2))) ) ).

% lexord_append_leftI
tff(fact_6439_append__replicate__commute,axiom,
    ! [A: $tType,N: nat,X: A,K: nat] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K),X)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K),X)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) ).

% append_replicate_commute
tff(fact_6440_replicate__add,axiom,
    ! [A: $tType,N: nat,M: nat,X: A] : aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M),X)) ).

% replicate_add
tff(fact_6441_remove1__append,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( remove1(A,X,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remove1(A,X,Xs)),Ys) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( remove1(A,X,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),remove1(A,X,Ys)) ) ) ) ).

% remove1_append
tff(fact_6442_enumerate__append__eq,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : enumerate(A,N,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(list(product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),append(product_prod(nat,A)),enumerate(A,N,Xs)),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_6443_append__eq__appendI,axiom,
    ! [A: $tType,Xs: list(A),Xs1: list(A),Zs: list(A),Ys: list(A),Us: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Xs1) = Zs )
     => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Us) )
       => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us) ) ) ) ).

% append_eq_appendI
tff(fact_6444_append__eq__append__conv2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),Ts: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Ts) )
    <=> ? [Us2: list(A)] :
          ( ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us2) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ys) = Ts ) )
          | ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2) = Zs )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ts) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_6445_eq__Nil__appendI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_6446_append_Oleft__neutral,axiom,
    ! [A: $tType,A2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),A2) = A2 ).

% append.left_neutral
tff(fact_6447_append__Nil,axiom,
    ! [A: $tType,Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) = Ys ).

% append_Nil
tff(fact_6448_remdups__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : remdups(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),remdups(A,Ys))) = remdups(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% remdups_append2
tff(fact_6449_comm__append__are__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
     => ? [M5: nat,N2: nat,Zs2: list(A)] :
          ( ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),M5),Zs2)) = Xs )
          & ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),N2),Zs2)) = Ys ) ) ) ).

% comm_append_are_replicate
tff(fact_6450_same__length__different,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != Ys )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => ? [Pre: list(A),X4: A,Xs4: list(A),Y2: A,Ys5: list(A)] :
            ( ( X4 != Y2 )
            & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))),Xs4)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))),Ys5)) ) ) ) ) ).

% same_length_different
tff(fact_6451_not__distinct__decomp,axiom,
    ! [A: $tType,Ws2: list(A)] :
      ( ~ distinct(A,Ws2)
     => ? [Xs2: list(A),Ys3: list(A),Zs2: list(A),Y2: A] : Ws2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))),Zs2)))) ) ).

% not_distinct_decomp
tff(fact_6452_not__distinct__conv__prefix,axiom,
    ! [A: $tType,As: list(A)] :
      ( ~ distinct(A,As)
    <=> ? [Xs3: list(A),Y4: A,Ys4: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),aa(list(A),set(A),set2(A),Xs3)))
          & distinct(A,Xs3)
          & ( As = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4)) ) ) ) ).

% not_distinct_conv_prefix
tff(fact_6453_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Ys: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),I),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),Ys) ) ) ).

% list_update_append1
tff(fact_6454_replicate__append__same,axiom,
    ! [A: $tType,I: nat,X: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),I),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),I),X)) ).

% replicate_append_same
tff(fact_6455_remdups__adj__append__two,axiom,
    ! [A: $tType,Xs: list(A),X: A,Y: A] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))))),if(list(A),aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ).

% remdups_adj_append_two
tff(fact_6456_remove1__split,axiom,
    ! [A: $tType,A2: A,Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Xs)))
     => ( ( remove1(A,A2,Xs) = Ys )
      <=> ? [Ls: list(A),Rs: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Rs)) )
            & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Ls)))
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls),Rs) ) ) ) ) ).

% remove1_split
tff(fact_6457_lexord__append__leftD,axiom,
    ! [A: $tType,X: list(A),U: list(A),V2: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V2))),lexord(A,R2)))
     => ( ! [A4: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4)),R2))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V2)),lexord(A,R2))) ) ) ).

% lexord_append_leftD
tff(fact_6458_lexord__append__rightI,axiom,
    ! [A: $tType,Y: list(A),X: list(A),R2: set(product_prod(A,A))] :
      ( ? [B9: A,Z4: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B9),Z4)
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),Y))),lexord(A,R2))) ) ).

% lexord_append_rightI
tff(fact_6459_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Qs))),lexord(A,R2)))
     => ( ( Xs != Ys )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
         => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
           => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R2))) ) ) ) ) ).

% lexord_sufE
tff(fact_6460_lex__append__leftD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X4: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R2))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R2)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R2))) ) ) ).

% lex_append_leftD
tff(fact_6461_lex__append__left__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X4: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R2))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R2)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R2))) ) ) ).

% lex_append_left_iff
tff(fact_6462_rotate1_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),rotate1(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% rotate1.simps(2)
tff(fact_6463_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R2)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs))),lex(A,R2))) ) ) ).

% lex_append_rightI
tff(fact_6464_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xs)),lenlex(A,R)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Vs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))),lenlex(A,R))) ) ) ).

% lenlex_append1
tff(fact_6465_nths__append,axiom,
    ! [A: $tType,L: list(A),L2: list(A),A3: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L2),A3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A3)),nths(A,L2,aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ade(list(A),fun(set(nat),fun(nat,bool)),L),A3)))) ).

% nths_append
tff(fact_6466_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_6467_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y4: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_6468_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(list(A)),subseqs(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),X)),aa(list(A),list(list(A)),subseqs(A),Xs))),aa(list(A),list(list(A)),subseqs(A),Xs)) ).

% subseqs.simps(2)
tff(fact_6469_nth__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N) = aa(nat,A,nth(A,Xs),N) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N) = aa(nat,A,nth(A,Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ) ).

% nth_append
tff(fact_6470_list__update__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A),X: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)),Ys) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),X)) ) ) ) ).

% list_update_append
tff(fact_6471_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2)))
     => ~ ! [X4: A,Y2: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y2)),R2))
           => ! [Us3: list(A),Vs3: list(A)] :
                ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Vs3)) )
               => ( Ys != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Vs3)) ) ) ) ) ).

% listrel1E
tff(fact_6472_listrel1I,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
       => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Vs)) )
         => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2))) ) ) ) ).

% listrel1I
tff(fact_6473_lexord__append__left__rightI,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),U: list(A),X: list(A),Y: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),X))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)))),lexord(A,R2))) ) ).

% lexord_append_left_rightI
tff(fact_6474_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R2)))
    <=> ( ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R2)) )
        | pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R2))) ) ) ).

% lexord_same_pref_iff
tff(fact_6475_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R2: set(product_prod(A,A)),V2: list(A),Z: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W2)),lexord(A,R2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),Z))),lexord(A,R2))) ) ) ).

% lexord_sufI
tff(fact_6476_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(list(B),list(product_prod(A,B))),map(B,product_prod(A,B)),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),Ys)),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_6477_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),listrel1(A,R2)))
    <=> ( ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R2)))
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_6478_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Ys))) ) ).

% horner_sum_append
tff(fact_6479_nths__Cons,axiom,
    ! [A: $tType,X: A,L: list(A),A3: set(nat)] : nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L),A3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),if(list(A),aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),nil(A))),nths(A,L,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_adf(set(nat),fun(nat,bool),A3)))) ).

% nths_Cons
tff(fact_6480_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
         => ? [N2: nat,Zs2: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N2))
              & ( aa(list(list(A)),list(A),concat(A),aa(list(A),list(list(A)),aa(nat,fun(list(A),list(list(A))),replicate(list(A)),N2),Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_6481_listrel1__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,R2) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_adg(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R2))) ).

% listrel1_def
tff(fact_6482_lexord__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lexord(A,R2) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_adh(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R2))) ).

% lexord_def
tff(fact_6483_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,I)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_6484_lex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_adi(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R2))) ).

% lex_conv
tff(fact_6485_nth__repl,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( M != N )
         => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),Xs)))),M) = aa(nat,A,nth(A,Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_6486_pos__n__replace,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),Xs)))) ) ) ).

% pos_n_replace
tff(fact_6487_drop0,axiom,
    ! [A: $tType,X2: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),zero_zero(nat)),X2) = X2 ).

% drop0
tff(fact_6488_drop__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),Xs) ).

% drop_drop
tff(fact_6489_drop__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) ).

% drop_Suc_Cons
tff(fact_6490_length__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_drop
tff(fact_6491_drop__update__cancel,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M),Xs) ) ) ).

% drop_update_cancel
tff(fact_6492_append__take__drop__id,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = Xs ).

% append_take_drop_id
tff(fact_6493_drop__replicate,axiom,
    ! [A: $tType,I: nat,K: nat,X: A] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),K),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I)),X) ).

% drop_replicate
tff(fact_6494_drop__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) = nil(A) ) ) ).

% drop_all
tff(fact_6495_drop__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) = nil(A) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_6496_drop__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_6497_drop__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),Ys)) ).

% drop_append
tff(fact_6498_drop__Cons__numeral,axiom,
    ! [A: $tType,V2: num,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(num,nat,numeral_numeral(nat),V2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat))),Xs) ).

% drop_Cons_numeral
tff(fact_6499_nth__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I)) ) ) ).

% nth_drop
tff(fact_6500_nth__via__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Y: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
     => ( aa(nat,A,nth(A,Xs),N) = Y ) ) ).

% nth_via_drop
tff(fact_6501_drop__zip,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(nat,fun(list(product_prod(A,B)),list(product_prod(A,B))),drop(product_prod(A,B)),N),zip(A,B,Xs,Ys)) = zip(A,B,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Ys)) ).

% drop_zip
tff(fact_6502_drop__Nil,axiom,
    ! [A: $tType,N: nat] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),nil(A)) = nil(A) ).

% drop_Nil
tff(fact_6503_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),zero_zero(nat)),Xs) = Xs ).

% drop_0
tff(fact_6504_in__set__dropD,axiom,
    ! [A: $tType,X: A,N: nat,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_dropD
tff(fact_6505_drop__take,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),M),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) ).

% drop_take
tff(fact_6506_take__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),Xs)) ).

% take_drop
tff(fact_6507_distinct__drop,axiom,
    ! [A: $tType,Xs: list(A),I: nat] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),Xs)) ) ).

% distinct_drop
tff(fact_6508_set__drop__subset,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_drop_subset
tff(fact_6509_drop__map,axiom,
    ! [A: $tType,B: $tType,N: nat,F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Xs)) ).

% drop_map
tff(fact_6510_drop__eq__nths,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),ord_less_eq(nat),N))) ).

% drop_eq_nths
tff(fact_6511_set__drop__subset__set__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M),Xs))),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_6512_drop__update__swap,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),M),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),X) ) ) ).

% drop_update_swap
tff(fact_6513_take__add,axiom,
    ! [A: $tType,I: nat,J: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),Xs))) ).

% take_add
tff(fact_6514_append__eq__conv__conj,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Zs )
    <=> ( ( Xs = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Xs)),Zs) )
        & ( Ys = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),Xs)),Zs) ) ) ) ).

% append_eq_conv_conj
tff(fact_6515_drop__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aTP_Lamp_adj(list(A),fun(nat,list(A)),Xs),N) ).

% drop_Cons
tff(fact_6516_nths__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I6: set(nat)] : nths(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs),I6) = nths(A,Xs,aa(set(nat),set(nat),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N)),I6)) ).

% nths_drop
tff(fact_6517_drop__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),Xs) ) ) ) ).

% drop_Cons'
tff(fact_6518_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),Xs_2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys_1),Ys_2) )
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( Xs_1 = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Xs_1)),Ys_1) )
            & ( Xs_2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),Xs_1)),Ys_1)),Ys_2) ) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),Ys_1)),Xs_1) = Ys_1 )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(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_6519_zip__append2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),zip(A,B,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(B),nat,size_size(list(B)),Ys)),Xs),Ys)),zip(A,B,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(B),nat,size_size(list(B)),Ys)),Xs),Zs)) ).

% zip_append2
tff(fact_6520_zip__append1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(A),Zs: list(B)] : zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),Zs) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),zip(A,B,Xs,aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),aa(list(A),nat,size_size(list(A)),Xs)),Zs))),zip(A,B,Ys,aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),aa(list(A),nat,size_size(list(A)),Xs)),Zs))) ).

% zip_append1
tff(fact_6521_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,I)),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_6522_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I: nat,J: nat] :
      ( distinct(A,Vs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Vs))),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),J),Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_6523_id__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,I)),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_6524_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A),A2: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),A2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,I)),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_6525_lexn__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_adk(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R2),N))) ).

% lexn_conv
tff(fact_6526_list_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F22: fun(A,fun(list(A),B))] : aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),nil(A)) = F1 ).

% list.simps(4)
tff(fact_6527_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_6528_list_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,fun(list(A),B)),List: list(A)] : aa(B,C,H,aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List)) = aa(list(A),C,aa(fun(A,fun(list(A),C)),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),C)),fun(list(A),C)),case_list(C,A),aa(B,C,H,F1)),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_adl(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),H),F22)),List) ).

% list.case_distrib
tff(fact_6529_list_Osimps_I5_J,axiom,
    ! [B: $tType,A: $tType,F1: B,F22: fun(A,fun(list(A),B)),X21: A,X222: list(A)] : aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(list(A),B,aa(A,fun(list(A),B),F22,X21),X222) ).

% list.simps(5)
tff(fact_6530_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N)))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ) ).

% lexn_length
tff(fact_6531_remdups__adj__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aTP_Lamp_adm(A,fun(A,fun(list(A),list(A))),X)),remdups_adj(A,Xs)) ).

% remdups_adj_Cons
tff(fact_6532_lex__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = aa(set(set(product_prod(list(A),list(A)))),set(product_prod(list(A),list(A))),complete_Sup_Sup(set(product_prod(list(A),list(A)))),aa(set(nat),set(set(product_prod(list(A),list(A)))),image2(nat,set(product_prod(list(A),list(A))),lexn(A,R2)),top_top(set(nat)))) ).

% lex_def
tff(fact_6533_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),X),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_adn(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) ).

% min_list.simps
tff(fact_6534_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_ado(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_adp(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_6535_transpose_Oelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( ( ( X = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
             => ( Y != transpose(A,Xss2) ) )
         => ~ ! [X4: A,Xs2: list(A),Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_ado(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_adp(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_6536_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> pp(aa(list(A),bool,aa(fun(A,fun(list(A),bool)),fun(list(A),bool),aa(bool,fun(fun(A,fun(list(A),bool)),fun(list(A),bool)),case_list(bool,A),fTrue),aTP_Lamp_adq(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(1)
tff(fact_6537_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> pp(aa(list(A),bool,aa(fun(A,fun(list(A),bool)),fun(list(A),bool),aa(bool,fun(fun(A,fun(list(A),bool)),fun(list(A),bool)),case_list(bool,A),fFalse),aTP_Lamp_adr(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(2)
tff(fact_6538_transpose_Osimps_I1_J,axiom,
    ! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).

% transpose.simps(1)
tff(fact_6539_transpose_Osimps_I2_J,axiom,
    ! [A: $tType,Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)) = transpose(A,Xss) ).

% transpose.simps(2)
tff(fact_6540_transpose__map__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(list(B))] : transpose(A,aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2)),Xs)) = aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2)),transpose(B,Xs)) ).

% transpose_map_map
tff(fact_6541_transpose__empty,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( transpose(A,Xs) = nil(list(A)) )
    <=> ! [X3: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
         => ( X3 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_6542_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(B),list(product_prod(A,B)),aa(fun(B,fun(list(B),list(product_prod(A,B)))),fun(list(B),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(fun(B,fun(list(B),list(product_prod(A,B)))),fun(list(B),list(product_prod(A,B)))),case_list(list(product_prod(A,B)),B),nil(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_ads(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs)),Ys) ).

% zip_Cons1
tff(fact_6543_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(A),list(product_prod(A,B)),aa(fun(A,fun(list(A),list(product_prod(A,B)))),fun(list(A),list(product_prod(A,B))),aa(list(product_prod(A,B)),fun(fun(A,fun(list(A),list(product_prod(A,B)))),fun(list(A),list(product_prod(A,B)))),case_list(list(product_prod(A,B)),A),nil(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_adt(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).

% zip_Cons
tff(fact_6544_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] :
      ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_ado(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_adp(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_6545_transpose_Opelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( accp(list(list(A)),transpose_rel(A),X)
       => ( ( ( X = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ accp(list(list(A)),transpose_rel(A),nil(list(A))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) ) )
           => ~ ! [X4: A,Xs2: list(A),Xss2: list(list(A))] :
                  ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_ado(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_adp(A,fun(list(A),list(list(A)))))),Xss2))))) )
                   => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Xss2)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_6546_transpose_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
     => ( transpose(A,nil(list(A))) = nil(list(A)) ) ) ).

% transpose.psimps(1)
tff(fact_6547_transpose_Opsimps_I2_J,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)) = transpose(A,Xss) ) ) ).

% transpose.psimps(2)
tff(fact_6548_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),bool)] :
      ( accp(list(list(A)),transpose_rel(A),A0)
     => ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
         => pp(aa(list(list(A)),bool,P,nil(list(A)))) )
       => ( ! [Xss2: list(list(A))] :
              ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))
             => ( pp(aa(list(list(A)),bool,P,Xss2))
               => pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))) ) )
         => ( ! [X4: A,Xs2: list(A),Xss2: list(list(A))] :
                ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Xss2))
               => ( pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_adp(A,fun(list(A),list(list(A)))))),Xss2)))))
                 => pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Xss2))) ) )
           => pp(aa(list(list(A)),bool,P,A0)) ) ) ) ) ).

% transpose.pinduct
tff(fact_6549_remdups__adj_Opelims,axiom,
    ! [A: $tType,X: list(A),Y: list(A)] :
      ( ( remdups_adj(A,X) = Y )
     => ( accp(list(A),remdups_adj_rel(A),X)
       => ( ( ( X = nil(A) )
           => ( ( Y = nil(A) )
             => ~ accp(list(A),remdups_adj_rel(A),nil(A)) ) )
         => ( ! [X4: A] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) )
               => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) )
                 => ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))) ) )
           => ~ ! [X4: A,Y2: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2)) )
                 => ( ( ( ( X4 = Y2 )
                       => ( Y = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)) ) )
                      & ( ( X4 != Y2 )
                       => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) )
                   => ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs2))) ) ) ) ) ) ) ).

% remdups_adj.pelims
tff(fact_6550_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list(int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),Js)) ) ) ) ).

% upto_aux_rec
tff(fact_6551_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))
       => ~ ( ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
               => ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
               => ( Y = nil(int) ) ) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)) ) ) ) ).

% upto.pelims
tff(fact_6552_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
         => ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
         => ( upto(I,J) = nil(int) ) ) ) ) ).

% upto.psimps
tff(fact_6553_upto__empty,axiom,
    ! [J: int,I: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
     => ( upto(I,J) = nil(int) ) ) ).

% upto_empty
tff(fact_6554_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil(int) = upto(I,J) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I)) ) ).

% upto_Nil2
tff(fact_6555_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil(int) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I)) ) ).

% upto_Nil
tff(fact_6556_upto__single,axiom,
    ! [I: int] : upto(I,I) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),nil(int)) ).

% upto_single
tff(fact_6557_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J))
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_6558_length__upto,axiom,
    ! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),I)),one_one(int))) ).

% length_upto
tff(fact_6559_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(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),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(1)
tff(fact_6560_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(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),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(2)
tff(fact_6561_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(3)
tff(fact_6562_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(4)
tff(fact_6563_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil(int)) ).

% upto_code
tff(fact_6564_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list(int)] : upto_aux(I,J,Js) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),Js) ).

% upto_aux_def
tff(fact_6565_distinct__upto,axiom,
    ! [I: int,J: int] : distinct(int,upto(I,J)) ).

% distinct_upto
tff(fact_6566_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1337092689740270186AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_6567_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_6568_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_6569_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6570_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_6571_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
         => ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
         => ( Y = nil(int) ) ) ) ) ).

% upto.elims
tff(fact_6572_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
       => ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
       => ( upto(I,J) = nil(int) ) ) ) ).

% upto.simps
tff(fact_6573_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_6574_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( upto(I,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),nil(int))) ) ) ).

% upto_rec2
tff(fact_6575_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_6576_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(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_6577_take__hd__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs))),nil(A))) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),Xs) ) ) ).

% take_hd_drop
tff(fact_6578_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] : aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(list(nat),list(bool),aa(fun(nat,bool),fun(list(nat),list(bool)),map(nat,bool),bit_se5641148757651400278ts_bit(A,A2)),upt(zero_zero(nat),N))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_6579_remdups__upt,axiom,
    ! [M: nat,N: nat] : remdups(nat,upt(M,N)) = upt(M,N) ).

% remdups_upt
tff(fact_6580_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( aa(list(nat),nat,hd(nat),upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_6581_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] : aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),drop(nat),M),upt(I,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M),J) ).

% drop_upt
tff(fact_6582_length__upt,axiom,
    ! [I: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) ).

% length_upt
tff(fact_6583_hd__remdups__adj,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),A,hd(A),remdups_adj(A,Xs)) = aa(list(A),A,hd(A),Xs) ).

% hd_remdups_adj
tff(fact_6584_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)),N))
     => ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),take(nat),M),upt(I,N)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)) ) ) ).

% take_upt
tff(fact_6585_hd__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_append2
tff(fact_6586_hd__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),A,hd(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = X ) ) ).

% hd_replicate
tff(fact_6587_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_6588_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or7035219750837199246ssThan(nat,M,N)) = upt(M,N) ).

% sorted_list_of_set_range
tff(fact_6589_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),J))
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),J),Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_take
tff(fact_6590_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_6591_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),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_6592_map__fst__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(nat),aa(fun(product_prod(nat,A),nat),fun(list(product_prod(nat,A)),list(nat)),map(product_prod(nat,A),nat),product_fst(nat,A)),enumerate(A,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_6593_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = aa(list(nat),list(nat),aa(nat,fun(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),N))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = nil(nat) ) ) ) ).

% upt_rec_numeral
tff(fact_6594_longest__common__prefix,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
    ? [Ps2: list(A),Xs4: list(A),Ys5: list(A)] :
      ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Xs4) )
      & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Ys5) )
      & ( ( Xs4 = nil(A) )
        | ( Ys5 = nil(A) )
        | ( aa(list(A),A,hd(A),Xs4) != aa(list(A),A,hd(A),Ys5) ) ) ) ).

% longest_common_prefix
tff(fact_6595_hd__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Ys) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Xs) ) ) ) ).

% hd_append
tff(fact_6596_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_6597_list_Oset__sel_I1_J,axiom,
    ! [A: $tType,A2: list(A)] :
      ( ( A2 != nil(A) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(list(A),A,hd(A),A2)),aa(list(A),set(A),set2(A),A2))) ) ).

% list.set_sel(1)
tff(fact_6598_hd__in__set,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(list(A),A,hd(A),Xs)),aa(list(A),set(A),set2(A),Xs))) ) ).

% hd_in_set
tff(fact_6599_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( aa(list(product_prod(A,B)),product_prod(A,B),hd(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(list(A),A,hd(A),Xs)),aa(list(B),B,hd(B),Ys)) ) ) ) ).

% hd_zip
tff(fact_6600_hd__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( Xs != nil(list(A)) )
     => ( ( aa(list(list(A)),list(A),hd(list(A)),Xs) != nil(A) )
       => ( aa(list(A),A,hd(A),aa(list(list(A)),list(A),concat(A),Xs)) = aa(list(A),A,hd(A),aa(list(list(A)),list(A),hd(list(A)),Xs)) ) ) ) ).

% hd_concat
tff(fact_6601_hd__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B)] :
      ( ( Xs != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs)) = aa(A,B,F2,aa(list(A),A,hd(A),Xs)) ) ) ).

% hd_map
tff(fact_6602_list_Omap__sel_I1_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F2: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),A2)) = aa(A,B,F2,aa(list(A),A,hd(A),A2)) ) ) ).

% list.map_sel(1)
tff(fact_6603_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or3652927894154168847AtMost(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),aa(nat,nat,suc,M))) ).

% greaterThanAtMost_upt
tff(fact_6604_atLeast__upt,axiom,
    ! [N: nat] : set_ord_lessThan(nat,N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),N)) ).

% atLeast_upt
tff(fact_6605_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or7035219750837199246ssThan(nat,I,J) = aa(list(nat),set(nat),set2(nat),upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_6606_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or1337092689740270186AtMost(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(N,aa(nat,nat,suc,M))) ).

% atLeastAtMost_upt
tff(fact_6607_map__add__upt,axiom,
    ! [N: nat,M: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),aTP_Lamp_adu(nat,fun(nat,nat),N)),upt(zero_zero(nat),M)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% map_add_upt
tff(fact_6608_distinct__upt,axiom,
    ! [I: nat,J: nat] : distinct(nat,upt(I,J)) ).

% distinct_upt
tff(fact_6609_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),suc),upt(M,N)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N)) ).

% map_Suc_upt
tff(fact_6610_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I: nat] : aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aTP_Lamp_adv(A,fun(nat,A),X)),upt(zero_zero(nat),I)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),I),X) ).

% map_replicate_trivial
tff(fact_6611_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : set_or5935395276787703475ssThan(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_6612_list_Osel_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),A,hd(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = X21 ).

% list.sel(1)
tff(fact_6613_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list(nat),Q4: nat] :
      ( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),M),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns)) = upt(M,Q4) )
    <=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns) = upt(aa(nat,nat,suc,M),Q4) ) ) ).

% upt_conv_Cons_Cons
tff(fact_6614_enumerate__map__upt,axiom,
    ! [A: $tType,N: nat,F2: fun(nat,A),M: nat] : enumerate(A,N,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(N,M))) = aa(list(nat),list(product_prod(nat,A)),aa(fun(nat,product_prod(nat,A)),fun(list(nat),list(product_prod(nat,A))),map(nat,product_prod(nat,A)),aTP_Lamp_adw(fun(nat,A),fun(nat,product_prod(nat,A)),F2)),upt(N,M)) ).

% enumerate_map_upt
tff(fact_6615_atMost__upto,axiom,
    ! [N: nat] : set_ord_atMost(nat,N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,N))) ).

% atMost_upto
tff(fact_6616_hd__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_6617_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_6618_enumerate__eq__zip,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,N,Xs) = zip(nat,A,upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_6619_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),N: nat] : aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F2,zero_zero(nat))),aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aTP_Lamp_adx(fun(nat,A),fun(nat,A),F2)),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_6620_map__decr__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),aTP_Lamp_jm(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_6621_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),nth(A,Xs)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_6622_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_6623_nth__map__upt,axiom,
    ! [A: $tType,I: nat,N: nat,M: nat,F2: fun(nat,A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(M,N))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_6624_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
        & ( I = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_6625_hd__drop__conv__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = aa(nat,A,nth(A,Xs),N) ) ) ).

% hd_drop_conv_nth
tff(fact_6626_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
       => ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
       => ( upt(I,J) = nil(nat) ) ) ) ).

% upt_rec
tff(fact_6627_enumerate__replicate__eq,axiom,
    ! [A: $tType,N: nat,M: nat,A2: A] : enumerate(A,N,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M),A2)) = aa(list(nat),list(product_prod(nat,A)),aa(fun(nat,product_prod(nat,A)),fun(list(nat),list(product_prod(nat,A))),map(nat,product_prod(nat,A)),aTP_Lamp_ady(A,fun(nat,product_prod(nat,A)),A2)),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% enumerate_replicate_eq
tff(fact_6628_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,M: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) )
     => ( ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
       => ( aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),F2),upt(M,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_6629_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => ( upt(I,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => ( upt(I,aa(nat,nat,suc,J)) = nil(nat) ) ) ) ).

% upt_Suc
tff(fact_6630_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( upt(I,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_6631_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 = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),A,hd(A),Xs)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_6632_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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)) = N ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),aa(fun(nat,list(A)),fun(list(nat),list(list(A))),map(nat,list(A)),aTP_Lamp_aea(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_6633_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & pp(aa(A,bool,P,Y))
        & ~ ? [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Ys)))
              & pp(aa(A,bool,P,X2)) ) ) ) ).

% extract_SomeE
tff(fact_6634_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & pp(aa(A,bool,P,Y))
        & ~ ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Ys)))
              & pp(aa(A,bool,P,X3)) ) ) ) ).

% extract_Some_iff
tff(fact_6635_extract__None__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( extract(A,P,Xs) = none(product_prod(list(A),product_prod(A,list(A)))) )
    <=> ~ ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(A,bool,P,X3)) ) ) ).

% extract_None_iff
tff(fact_6636_extract__Nil__code,axiom,
    ! [A: $tType,P: fun(A,bool)] : extract(A,P,nil(A)) = none(product_prod(list(A),product_prod(A,list(A)))) ).

% extract_Nil_code
tff(fact_6637_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),nil(A)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),X),Xs))) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A))),none(product_prod(list(A),product_prod(A,list(A)))),aa(fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_aec(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),X)),extract(A,P,Xs)) ) ) ) ).

% extract_Cons_code
tff(fact_6638_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A),F2: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),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_aed(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F2)),X6) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_6639_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),aTP_Lamp_aee(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_aef(nat,fun(list(A),bool),I)),Xs)) ) ) ).

% nth_transpose
tff(fact_6640_filter__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),Q),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_aeg(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)),Xs) ).

% filter_filter
tff(fact_6641_filter__True,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) )
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = Xs ) ) ).

% filter_True
tff(fact_6642_filter__append,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys)) ).

% filter_append
tff(fact_6643_remove1__filter__not,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ~ pp(aa(A,bool,P,X))
     => ( remove1(A,X,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ) ) ).

% remove1_filter_not
tff(fact_6644_removeAll__filter__not,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ~ pp(aa(A,bool,P,X))
     => ( aa(list(A),list(A),removeAll(A,X),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ) ) ).

% removeAll_filter_not
tff(fact_6645_set__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_aeh(fun(A,bool),fun(list(A),fun(A,bool)),P),Xs)) ).

% set_filter
tff(fact_6646_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_6647_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) )
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Ns)))
             => ( X3 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_6648_filter__False,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(A,bool,P,X4)) )
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = nil(A) ) ) ).

% filter_False
tff(fact_6649_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F2: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(fun(B,A),fun(B,bool),comp(A,bool,B,P),F2)),Xs)) ).

% length_filter_map
tff(fact_6650_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aTP_Lamp_aei(B,A)),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_6651_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(M,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_co(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% sum_list_upt
tff(fact_6652_empty__filter__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) )
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(A,bool,P,X3)) ) ) ).

% empty_filter_conv
tff(fact_6653_filter__empty__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = nil(A) )
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(A,bool,P,X3)) ) ) ).

% filter_empty_conv
tff(fact_6654_filter__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ( Xs = Ys )
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ys)))
           => ( pp(aa(A,bool,P,X4))
            <=> pp(aa(A,bool,Q,X4)) ) )
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),Q),Ys) ) ) ) ).

% filter_cong
tff(fact_6655_filter__id__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = Xs )
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) ) ) ).

% filter_id_conv
tff(fact_6656_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_bu(fun(A,bool),fun(A,bool),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_6657_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),fequal(A),X)),Xs))),X) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),fequal(A),X)),Xs) ).

% replicate_length_filter
tff(fact_6658_filter__set,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : filter3(A,P,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) ).

% filter_set
tff(fact_6659_inter__set__filter,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_a(set(A),fun(A,bool),A3)),Xs)) ).

% inter_set_filter
tff(fact_6660_distinct__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) ) ).

% distinct_filter
tff(fact_6661_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_filter_le
tff(fact_6662_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% filter_is_subset
tff(fact_6663_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).

% member_le_sum_list
tff(fact_6664_filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(fun(B,A),fun(B,bool),comp(A,bool,B,P),F2)),Xs)) ).

% filter_map
tff(fact_6665_sum__list__map__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B),P: fun(B,bool),F2: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(list(B),set(B),set2(B),Xs)))
             => ( ~ pp(aa(B,bool,P,X4))
               => ( aa(B,A,F2,X4) = zero_zero(A) ) ) )
         => ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_6666_length__concat,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(B),nat,size_size(list(B)),aa(list(list(B)),list(B),concat(B),Xss)) = aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(list(B)),list(nat),aa(fun(list(B),nat),fun(list(list(B)),list(nat)),map(list(B),nat),size_size(list(B))),Xss)) ).

% length_concat
tff(fact_6667_filter__concat,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(list(A))] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P2),aa(list(list(A)),list(A),concat(A),Xs)) = aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P2)),Xs)) ).

% filter_concat
tff(fact_6668_distinct__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),P: fun(B,bool)] :
      ( distinct(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
     => distinct(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs))) ) ).

% distinct_map_filter
tff(fact_6669_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F2: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_aej(fun(B,A),fun(fun(B,bool),fun(B,A)),F2),P)),Xs)) ) ).

% sum_list_map_filter'
tff(fact_6670_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),F2),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)))),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),F2),Xs)))) ).

% sum_list_filter_le_nat
tff(fact_6671_remdups__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : remdups(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),remdups(A,Xs)) ).

% remdups_filter
tff(fact_6672_filter_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),nil(A)) = nil(A) ).

% filter.simps(1)
tff(fact_6673_filter__replicate,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,N: nat] :
      ( ( pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = nil(A) ) ) ) ).

% filter_replicate
tff(fact_6674_removeAll__filter__not__eq,axiom,
    ! [A: $tType,X: A] : removeAll(A,X) = aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_aek(A,fun(A,bool),X)) ).

% removeAll_filter_not_eq
tff(fact_6675_partition__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] : pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),shuffles(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_bu(fun(A,bool),fun(A,bool),P)),Xs)))) ).

% partition_in_shuffles
tff(fact_6676_filter__insort__triv,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),X: B,F2: fun(B,A),Xs: list(B)] :
          ( ~ pp(aa(B,bool,P,X))
         => ( aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs) ) ) ) ).

% filter_insort_triv
tff(fact_6677_filter__remove1,axiom,
    ! [A: $tType,Q: fun(A,bool),X: A,Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),Q),remove1(A,X,Xs)) = remove1(A,X,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),Q),Xs)) ).

% filter_remove1
tff(fact_6678_filter__shuffles,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A)] : aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P)),shuffles(A,Xs,Ys)) = shuffles(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys)) ).

% filter_shuffles
tff(fact_6679_filter_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ) ) ) ).

% filter.simps(2)
tff(fact_6680_length__filter__less,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P,X))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% length_filter_less
tff(fact_6681_Cons__eq__filterD,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys) )
     => ? [Us3: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs3)) )
          & ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Us3)))
             => ~ pp(aa(A,bool,P,X2)) )
          & pp(aa(A,bool,P,X))
          & ( Xs = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Vs3) ) ) ) ).

% Cons_eq_filterD
tff(fact_6682_filter__eq__ConsD,axiom,
    ! [A: $tType,P: fun(A,bool),Ys: list(A),X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
     => ? [Us3: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs3)) )
          & ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Us3)))
             => ~ pp(aa(A,bool,P,X2)) )
          & pp(aa(A,bool,P,X))
          & ( Xs = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Vs3) ) ) ) ).

% filter_eq_ConsD
tff(fact_6683_Cons__eq__filter__iff,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys) )
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs2)) )
          & ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Us2)))
             => ~ pp(aa(A,bool,P,X3)) )
          & pp(aa(A,bool,P,X))
          & ( Xs = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Vs2) ) ) ) ).

% Cons_eq_filter_iff
tff(fact_6684_filter__eq__Cons__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Ys: list(A),X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs2)) )
          & ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Us2)))
             => ~ pp(aa(A,bool,P,X3)) )
          & pp(aa(A,bool,P,X))
          & ( Xs = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Vs2) ) ) ) ).

% filter_eq_Cons_iff
tff(fact_6685_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),aTP_Lamp_ael(fun(A,B),fun(A,fun(A,bool)),F2),Y)),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),fequal(A),Y)),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_6686_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X4)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_6687_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X4)) )
         => ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
          <=> ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
               => ( X3 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_6688_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A))) ) ) ).

% sum_list_nonpos
tff(fact_6689_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),abs_abs(A)),Xs)))) ) ).

% sum_list_abs
tff(fact_6690_filter__in__nths,axiom,
    ! [A: $tType,Xs: list(A),S: set(nat)] :
      ( distinct(A,Xs)
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(set(nat),fun(A,bool),aTP_Lamp_aem(list(A),fun(set(nat),fun(A,bool)),Xs),S)),Xs) = nths(A,Xs,S) ) ) ).

% filter_in_nths
tff(fact_6691_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X4)),aa(A,B,G,X4))) )
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xs)))) ) ) ).

% sum_list_mono
tff(fact_6692_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aen(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_6693_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(A,fun(A,bool),aTP_Lamp_aeo(A,fun(A,bool)),Y)),Xs)) ).

% set_minus_filter_out
tff(fact_6694_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_aep(list(A),fun(A,bool),Ys)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_6695_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_aeq(list(A),fun(A,bool),Ys)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_6696_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_aep(list(A),fun(A,bool),Xs)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_6697_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_aeq(list(A),fun(A,bool),Xs)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_6698_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(list(A),fun(nat,bool),aTP_Lamp_aer(fun(A,bool),fun(list(A),fun(nat,bool)),P),Xs))) ).

% filter_eq_nths
tff(fact_6699_length__filter__conv__card,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P2),Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(list(A),fun(nat,bool),aTP_Lamp_aer(fun(A,bool),fun(list(A),fun(nat,bool)),P2),Xs))) ).

% length_filter_conv_card
tff(fact_6700_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),aa(list(A),A,groups8242544230860333062m_list(A),Ns))) ) ) ).

% elem_le_sum_list
tff(fact_6701_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,X4)),aa(A,B,G,X4))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xs)))) ) ) ) ).

% sum_list_strict_mono
tff(fact_6702_sum_Odistinct__set__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(B),G: fun(B,A)] :
          ( distinct(B,Xs)
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs)) ) ) ) ).

% sum.distinct_set_conv_list
tff(fact_6703_sum__list__distinct__conv__sum__set,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Xs: list(B),F2: fun(B,C)] :
          ( distinct(B,Xs)
         => ( aa(list(C),C,groups8242544230860333062m_list(C),aa(list(B),list(C),aa(fun(B,C),fun(list(B),list(C)),map(B,C),F2),Xs)) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),F2),aa(list(B),set(B),set2(B),Xs)) ) ) ) ).

% sum_list_distinct_conv_sum_set
tff(fact_6704_sum__list__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [X: B,Xs: list(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(list(B),set(B),set2(B),Xs)))
         => ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,X)),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),remove1(B,X,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_6705_sum__code,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),remdups(B,Xs))) ) ).

% sum_code
tff(fact_6706_size__list__conv__sum__list,axiom,
    ! [B: $tType,F2: fun(B,nat),Xs: list(B)] : aa(list(B),nat,size_list(B,F2),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(B),list(nat),aa(fun(B,nat),fun(list(B),list(nat)),map(B,nat),F2),Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% size_list_conv_sum_list
tff(fact_6707_sum__list__triv,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [R2: B,Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),aTP_Lamp_aes(B,fun(C,B),R2)),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(list(C),nat,size_size(list(C)),Xs))),R2) ) ).

% sum_list_triv
tff(fact_6708_sum__list__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),aTP_Lamp_aet(fun(A,nat),fun(A,nat),F2)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),F2),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_6709_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,bool),set(A),collect(A),P)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_6710_sum__list__sum__nth,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),Xs) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),nth(B,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_6711_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_aeu(nat,fun(nat,fun(list(nat),bool)),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),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_aev(nat,fun(nat,fun(list(nat),bool)),M),N4)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_aew(nat,fun(nat,fun(list(nat),bool)),M),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_6712_card__length__sum__list,axiom,
    ! [M: nat,N4: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_aeu(nat,fun(nat,fun(list(nat),bool)),M),N4))) = binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),M)),one_one(nat)),N4) ).

% card_length_sum_list
tff(fact_6713_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),aa(fun(A,nat),fun(list(A),list(nat)),map(A,nat),F2),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_aex(fun(A,nat),fun(list(A),fun(A,nat)),F2),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_6714_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( aa(list(A),A,groups8242544230860333062m_list(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),X)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_6715_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_ado(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),hd(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_aey(list(A),bool)),Xss)) ).

% transpose_aux_filter_head
tff(fact_6716_map__filter__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F2,Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),aa(fun(A,option(B)),fun(A,B),comp(option(B),B,A,the2(B)),F2)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_abr(fun(A,option(B)),fun(A,bool),F2)),Xs)) ).

% map_filter_def
tff(fact_6717_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(list(A))),list(list(A)),concat(list(A)),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A))),aa(list(list(A)),fun(fun(A,fun(list(A),list(list(A)))),fun(list(A),list(list(A)))),case_list(list(list(A)),A),nil(list(A))),aTP_Lamp_adp(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),tl(A)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_aey(list(A),bool)),Xss)) ).

% transpose_aux_filter_tail
tff(fact_6718_tl__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),tl(nat),upt(M,N)) = upt(aa(nat,nat,suc,M),N) ).

% tl_upt
tff(fact_6719_tl__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xs)),Ys) ) ) ).

% tl_append2
tff(fact_6720_remdups__adj__Cons__alt,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),tl(A),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))) = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ).

% remdups_adj_Cons_alt
tff(fact_6721_length__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_tl
tff(fact_6722_hd__Cons__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),aa(list(A),list(A),tl(A),Xs)) = Xs ) ) ).

% hd_Cons_tl
tff(fact_6723_list_Ocollapse,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) = List ) ) ).

% list.collapse
tff(fact_6724_tl__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),list(A),tl(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),X) ).

% tl_replicate
tff(fact_6725_list_Omap__sel_I2_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F2: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),list(B),tl(B),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),A2)) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),aa(list(A),list(A),tl(A),A2)) ) ) ).

% list.map_sel(2)
tff(fact_6726_map__tl,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),tl(B),Xs)) = aa(list(A),list(A),tl(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) ).

% map_tl
tff(fact_6727_list_Osel_I3_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = X222 ).

% list.sel(3)
tff(fact_6728_map__filter__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,option(A))] : map_filter(B,A,F2,nil(B)) = nil(A) ).

% map_filter_simps(2)
tff(fact_6729_list_Osel_I2_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).

% list.sel(2)
tff(fact_6730_tl__Nil,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ? [X3: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) ) ) ).

% tl_Nil
tff(fact_6731_Nil__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),tl(A),Xs) )
    <=> ( ( Xs = nil(A) )
        | ? [X3: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) ) ) ).

% Nil_tl
tff(fact_6732_tl__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),tl(A),Xs)) ).

% tl_drop
tff(fact_6733_drop__Suc,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,suc,N)),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),tl(A),Xs)) ).

% drop_Suc
tff(fact_6734_take__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),tl(A),Xs)) = aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),Xs)) ).

% take_tl
tff(fact_6735_distinct__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),tl(A),Xs)) ) ).

% distinct_tl
tff(fact_6736_list_Oexpand,axiom,
    ! [A: $tType,List: list(A),List2: list(A)] :
      ( ( ( List = nil(A) )
      <=> ( List2 = nil(A) ) )
     => ( ( ( List != nil(A) )
         => ( ( List2 != nil(A) )
           => ( ( aa(list(A),A,hd(A),List) = aa(list(A),A,hd(A),List2) )
              & ( aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),tl(A),List2) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
tff(fact_6737_list_Oset__sel_I2_J,axiom,
    ! [A: $tType,A2: list(A),X: A] :
      ( ( A2 != nil(A) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),A2))))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),A2))) ) ) ).

% list.set_sel(2)
tff(fact_6738_map__of__filter__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,Z: A,P: fun(B,fun(A,bool))] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Z) )
     => ( pp(aa(A,bool,aa(B,fun(A,bool),P,K),Z))
       => ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(fun(product_prod(B,A),bool),fun(list(product_prod(B,A)),list(product_prod(B,A))),filter2(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P)),Xs)),K) = aa(A,option(A),some(A),Z) ) ) ) ).

% map_of_filter_in
tff(fact_6739_tl__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),nil(A)),aTP_Lamp_aez(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_6740_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,fun(list(A),list(A))),fun(list(A),list(A)),aa(list(A),fun(fun(A,fun(list(A),list(A))),fun(list(A),list(A))),case_list(list(A),A),aa(list(A),list(A),tl(A),Ys)),aTP_Lamp_afa(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_6741_list_Oexhaust__sel,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) ) ) ).

% list.exhaust_sel
tff(fact_6742_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),X: B,Xs: list(B)] : map_filter(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = case_option(list(A),A,map_filter(B,A,F2,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_afb(fun(B,option(A)),fun(list(B),fun(A,list(A))),F2),Xs),aa(B,option(A),F2,X)) ).

% map_filter_simps(1)
tff(fact_6743_tl__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(list(A),list(A),tl(A),Xs)) ).

% tl_take
tff(fact_6744_list_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,List: list(A),F1: B,F22: fun(A,fun(list(A),B))] :
      ( ( ( List = nil(A) )
       => ( aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List) = F1 ) )
      & ( ( List != nil(A) )
       => ( aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List) = aa(list(A),B,aa(A,fun(list(A),B),F22,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) ) ) ) ).

% list.case_eq_if
tff(fact_6745_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))) ) ) ) ).

% Nitpick.size_list_simp(2)
tff(fact_6746_nth__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).

% nth_tl
tff(fact_6747_remdups__adj__append,axiom,
    ! [A: $tType,Xs_1: list(A),X: A,Xs_2: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs_2))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))))),aa(list(A),list(A),tl(A),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs_2)))) ).

% remdups_adj_append
tff(fact_6748_Cons__in__shuffles__iff,axiom,
    ! [A: $tType,Z: A,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Zs)),shuffles(A,Xs,Ys)))
    <=> ( ( ( Xs != nil(A) )
          & ( aa(list(A),A,hd(A),Xs) = Z )
          & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,aa(list(A),list(A),tl(A),Xs),Ys))) )
        | ( ( Ys != nil(A) )
          & ( aa(list(A),A,hd(A),Ys) = Z )
          & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,aa(list(A),list(A),tl(A),Ys)))) ) ) ) ).

% Cons_in_shuffles_iff
tff(fact_6749_list_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,fun(list(A),B)),List: list(A)] :
      ( pp(aa(B,bool,P,aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List)))
    <=> ( ( ( List = nil(A) )
         => pp(aa(B,bool,P,F1)) )
        & ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) )
         => pp(aa(B,bool,P,aa(list(A),B,aa(A,fun(list(A),B),F22,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)))) ) ) ) ).

% list.split_sel
tff(fact_6750_list_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,fun(list(A),B)),List: list(A)] :
      ( pp(aa(B,bool,P,aa(list(A),B,aa(fun(A,fun(list(A),B)),fun(list(A),B),aa(B,fun(fun(A,fun(list(A),B)),fun(list(A),B)),case_list(B,A),F1),F22),List)))
    <=> ~ ( ( ( List = nil(A) )
            & ~ pp(aa(B,bool,P,F1)) )
          | ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) )
            & ~ pp(aa(B,bool,P,aa(list(A),B,aa(A,fun(list(A),B),F22,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)))) ) ) ) ).

% list.split_sel_asm
tff(fact_6751_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P: fun(nat,bool),Xs: list(A),Is: list(nat)] : aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_afc(fun(nat,bool),fun(product_prod(A,nat),bool),P)),zip(A,nat,Xs,Is))) = aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_afd(fun(nat,bool),fun(product_prod(A,nat),bool),P)),zip(A,nat,Xs,aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),map(nat,nat),suc),Is)))) ).

% nths_shift_lemma_Suc
tff(fact_6752_take__Suc,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,suc,N)),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),tl(A),Xs))) ) ) ).

% take_Suc
tff(fact_6753_rotate1__hd__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),rotate1(A),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),nil(A))) ) ) ).

% rotate1_hd_tl
tff(fact_6754_nths__shift__lemma,axiom,
    ! [A: $tType,A3: set(nat),Xs: list(A),I: nat] : aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_afe(set(nat),fun(product_prod(A,nat),bool),A3)),zip(A,nat,Xs,upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_aff(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A3),I)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_6755_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),nat,size_list(A,F2),Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),nat,size_list(A,F2),Xs) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,aa(list(A),A,hd(A),Xs))),aa(list(A),nat,size_list(A,F2),aa(list(A),list(A),tl(A),Xs)))) ) ) ) ).

% Nitpick.size_list_simp(1)
tff(fact_6756_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat)] : nths(A,Xs,A3) = aa(list(product_prod(A,nat)),list(A),aa(fun(product_prod(A,nat),A),fun(list(product_prod(A,nat)),list(A)),map(product_prod(A,nat),A),product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),aa(fun(product_prod(A,nat),bool),fun(list(product_prod(A,nat)),list(product_prod(A,nat))),filter2(product_prod(A,nat)),aTP_Lamp_afe(set(nat),fun(product_prod(A,nat),bool),A3)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_6757_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_afg(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F2),P),Xs) ).

% map_filter_map_filter
tff(fact_6758_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,aa(list(list(A)),fun(nat,nat),aa(fun(list(A),fun(nat,nat)),fun(list(list(A)),fun(nat,nat)),foldr(list(A),nat),aTP_Lamp_afh(list(A),fun(nat,nat))),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_aey(list(A),bool)),Xs)) ).

% transpose_max_length
tff(fact_6759_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),aa(nat,nat,aa(list(list(B)),fun(nat,nat),aa(fun(list(B),fun(nat,nat)),fun(list(list(B)),fun(nat,nat)),foldr(list(B),nat),aTP_Lamp_afi(list(B),fun(nat,nat))),Xss),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,aa(list(list(B)),fun(nat,nat),aa(fun(list(B),fun(nat,nat)),fun(list(list(B)),fun(nat,nat)),foldr(list(B),nat),aTP_Lamp_afj(list(B),fun(nat,nat))),aa(list(list(B)),list(list(B)),aa(fun(list(B),bool),fun(list(list(B)),list(list(B))),filter2(list(B)),aTP_Lamp_afk(list(B),bool)),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_6760_foldr__append,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Xs: list(B),Ys: list(B),A2: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),A2) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),Xs),aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),Ys),A2)) ).

% foldr_append
tff(fact_6761_foldr__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),N: nat,X: B] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),X)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(B,fun(A,A),F2,X)) ).

% foldr_replicate
tff(fact_6762_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list(B),K: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A4: A,X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(list(B),set(B),set2(B),L)))
             => ( aa(A,A,aa(B,fun(A,A),F2,X4),A4) = aa(A,A,aa(B,fun(A,A),G,X4),A4) ) )
         => ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),L),A2) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),G),K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_6763_foldr__Cons,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),X: A,Xs: list(A)] : aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F2),Xs)) ).

% foldr_Cons
tff(fact_6764_foldr__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F2: fun(C,B),Xs: list(C),A2: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),G),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F2),Xs)),A2) = aa(A,A,aa(list(C),fun(A,A),aa(fun(C,fun(A,A)),fun(list(C),fun(A,A)),foldr(C,A),aa(fun(C,B),fun(C,fun(A,A)),comp(B,fun(A,A),C,G),F2)),Xs),A2) ).

% foldr_map
tff(fact_6765_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),foldr(A,A),plus_plus(A)),Xs),zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_6766_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A2),Xs) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_afl(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A2)),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_6767_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(nat,nat,aa(list(list(A)),fun(nat,nat),aa(fun(list(A),fun(nat,nat)),fun(list(list(A)),fun(nat,nat)),foldr(list(A),nat),aTP_Lamp_afh(list(A),fun(nat,nat))),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_6768_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [F2: fun(nat,B),Ns: list(nat)] :
          ( ! [X4: nat,Y2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Y2))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(nat,B,F2,X4)),aa(nat,B,F2,Y2))) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(nat),list(B),aa(fun(nat,B),fun(list(nat),list(B)),map(nat,B),F2),Ns)))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_6769_length__product__lists,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),aa(list(list(B)),list(list(B)),product_lists(B),Xss)) = aa(nat,nat,aa(list(nat),fun(nat,nat),aa(fun(nat,fun(nat,nat)),fun(list(nat),fun(nat,nat)),foldr(nat,nat),times_times(nat)),aa(list(list(B)),list(nat),aa(fun(list(B),nat),fun(list(list(B)),list(nat)),map(list(B),nat),size_size(list(B))),Xss)),one_one(nat)) ).

% length_product_lists
tff(fact_6770_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,bool)),F2: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_afm(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R),F2),Xs) ) ).

% sorted_wrt_map
tff(fact_6771_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
        <=> sorted_wrt(B,aTP_Lamp_afn(fun(B,A),fun(B,fun(B,bool)),F2),Xs) ) ) ).

% sorted_map
tff(fact_6772_sorted__wrt__filter,axiom,
    ! [A: $tType,F2: fun(A,fun(A,bool)),Xs: list(A),P: fun(A,bool)] :
      ( sorted_wrt(A,F2,Xs)
     => sorted_wrt(A,F2,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) ) ).

% sorted_wrt_filter
tff(fact_6773_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(list(A),fun(A,bool),aTP_Lamp_afo(fun(list(A),A),fun(list(A),fun(A,bool)),G),Xs)),Xs)) ) ).

% sorted_same
tff(fact_6774_sorted__tl,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),tl(A),Xs)) ) ) ).

% sorted_tl
tff(fact_6775_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_6776_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
               => ! [Xa3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(list(A),set(A),set2(A),Ys)))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa3)) ) ) ) ) ) ).

% sorted_append
tff(fact_6777_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : sorted_wrt(nat,ord_less(nat),upt(M,N)) ).

% sorted_wrt_upt
tff(fact_6778_sorted__upt,axiom,
    ! [M: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,N)) ).

% sorted_upt
tff(fact_6779_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_6780_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I6: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I6)) ) ) ).

% sorted_nths
tff(fact_6781_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_6782_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_6783_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A2,Xs)) ) ) ).

% sorted_remove1
tff(fact_6784_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_6785_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: nat,X: A] : sorted_wrt(A,ord_less_eq(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) ) ).

% sorted_replicate
tff(fact_6786_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),X),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_6787_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_afp(A,fun(A,bool)),Xs) ).

% sorted_wrt_true
tff(fact_6788_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_6789_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_6790_sorted__wrt_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : sorted_wrt(A,P,nil(A)) ).

% sorted_wrt.simps(1)
tff(fact_6791_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_6792_sorted__wrt1,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A] : sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% sorted_wrt1
tff(fact_6793_sorted__wrt__drop,axiom,
    ! [A: $tType,F2: fun(A,fun(A,bool)),Xs: list(A),N: nat] :
      ( sorted_wrt(A,F2,Xs)
     => sorted_wrt(A,F2,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) ) ).

% sorted_wrt_drop
tff(fact_6794_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) ) ) ).

% sorted_drop
tff(fact_6795_sorted__wrt__take,axiom,
    ! [A: $tType,F2: fun(A,fun(A,bool)),Xs: list(A),N: nat] :
      ( sorted_wrt(A,F2,Xs)
     => sorted_wrt(A,F2,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) ) ).

% sorted_wrt_take
tff(fact_6796_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) ) ) ).

% sorted_take
tff(fact_6797_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_6798_sorted__wrt__append,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( sorted_wrt(A,P,Xs)
        & sorted_wrt(A,P,Ys)
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
           => ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(list(A),set(A),set2(A),Ys)))
               => pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Xa3)) ) ) ) ) ).

% sorted_wrt_append
tff(fact_6799_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys)
           => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_6800_sorted__wrt__mono__rel,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool)),Q: fun(A,fun(A,bool))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),aa(list(A),set(A),set2(A),Xs)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Y2))
             => pp(aa(A,bool,aa(A,fun(A,bool),Q,X4),Y2)) ) ) )
     => ( sorted_wrt(A,P,Xs)
       => sorted_wrt(A,Q,Xs) ) ) ).

% sorted_wrt_mono_rel
tff(fact_6801_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
        <=> ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Ys)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X3)) )
            & sorted_wrt(A,ord_less(A),Ys) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_6802_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
        <=> ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Ys)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X3)) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_6803_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ).

% sorted1
tff(fact_6804_sorted__wrt_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Ys: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
    <=> ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Ys)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,X),X3)) )
        & sorted_wrt(A,P,Ys) ) ) ).

% sorted_wrt.simps(2)
tff(fact_6805_sorted__wrt_Oelims_I3_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A)] :
      ( ~ sorted_wrt(A,X,Xa2)
     => ~ ! [X4: A,Ys3: list(A)] :
            ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3) )
           => ( ! [Xa4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),aa(list(A),set(A),set2(A),Ys3)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),X,X4),Xa4)) )
              & sorted_wrt(A,X,Ys3) ) ) ) ).

% sorted_wrt.elims(3)
tff(fact_6806_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_6807_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),I: nat,J: nat] :
      ( sorted_wrt(A,P,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_6808_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys)
             => ( distinct(A,Ys)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                 => ( Xs = Ys ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_6809_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => sorted_wrt(A,P,Xs) ) ).

% sorted_wrt01
tff(fact_6810_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs))) ) ) ).

% sorted_filter
tff(fact_6811_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) ) ) ).

% sorted_insort_key
tff(fact_6812_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),remove1(B,X,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_6813_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),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_afq(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F2),G),Xs)),Xs))) ) ).

% sorted_map_same
tff(fact_6814_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] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_6815_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_6816_sorted__wrt_Oelims_I2_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A)] :
      ( sorted_wrt(A,X,Xa2)
     => ( ( Xa2 != nil(A) )
       => ~ ! [X4: A,Ys3: list(A)] :
              ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3) )
             => ~ ( ! [Xa: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),aa(list(A),set(A),set2(A),Ys3)))
                     => pp(aa(A,bool,aa(A,fun(A,bool),X,X4),Xa)) )
                  & sorted_wrt(A,X,Ys3) ) ) ) ) ).

% sorted_wrt.elims(2)
tff(fact_6817_sorted__wrt_Oelims_I1_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Y: bool] :
      ( ( sorted_wrt(A,X,Xa2)
      <=> pp(Y) )
     => ( ( ( Xa2 = nil(A) )
         => ~ pp(Y) )
       => ~ ! [X4: A,Ys3: list(A)] :
              ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3) )
             => ( pp(Y)
              <=> ~ ( ! [Xa3: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(list(A),set(A),set2(A),Ys3)))
                       => pp(aa(A,bool,aa(A,fun(A,bool),X,X4),Xa3)) )
                    & sorted_wrt(A,X,Ys3) ) ) ) ) ) ).

% sorted_wrt.elims(1)
tff(fact_6818_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ? [X4: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X4) = A3 )
              & sorted_wrt(A,ord_less_eq(A),X4)
              & distinct(A,X4)
              & ! [Y3: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y3) = A3 )
                    & sorted_wrt(A,ord_less_eq(A),Y3)
                    & distinct(A,Y3) )
                 => ( Y3 = X4 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_6819_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_6820_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,bool),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
         => ( pp(aa(B,bool,P,X))
           => ( aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs)) ) ) ) ) ).

% filter_insort
tff(fact_6821_concat__conv__foldr,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),list(A),concat(A),Xss) = aa(list(A),list(A),aa(list(list(A)),fun(list(A),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(list(A)),fun(list(A),list(A))),foldr(list(A),list(A)),append(A)),Xss),nil(A)) ).

% concat_conv_foldr
tff(fact_6822_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_6823_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4)))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_6824_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_6825_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).

% sorted_nth_mono
tff(fact_6826_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ~ ! [L3: list(A)] :
                ( sorted_wrt(A,ord_less(A),L3)
               => ( ( aa(list(A),set(A),set2(A),L3) = A3 )
                 => ( aa(list(A),nat,size_size(list(A)),L3) != aa(set(A),nat,finite_card(A),A3) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_6827_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_6828_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
                & pp(aa(A,bool,P,X2)) )
           => ( find(A,P,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_afr(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_6829_sorted__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),aa(fun(product_prod(nat,A),nat),fun(list(product_prod(nat,A)),list(nat)),map(product_prod(nat,A),nat),product_fst(nat,A)),enumerate(A,N,Xs))) ).

% sorted_enumerate
tff(fact_6830_map__sorted__distinct__set__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Ys: list(B)] :
          ( inj_on(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)))
         => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
           => ( distinct(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
             => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys))
               => ( distinct(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Ys))
                 => ( ( aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),set2(B),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_6831_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),L: list(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A3 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A3) ) )
          <=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_6832_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),A2)) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),A2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_6833_insort__key__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [A2: B,Xs: list(B),F2: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),aa(list(B),set(B),set2(B),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
           => ( ( aa(list(B),B,hd(B),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_afs(B,fun(fun(B,A),fun(B,bool)),A2),F2)),Xs)) = A2 )
             => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),A2),remove1(B,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_6834_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_aef(nat,fun(list(A),bool),I)),Xs))))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_6835_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),list(A),aa(fun(list(A),A),fun(list(list(A)),list(A)),map(list(A),A),aTP_Lamp_aee(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_aef(nat,fun(list(A),bool),I)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_6836_rev__is__rev__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),rev(A),Ys) )
    <=> ( Xs = Ys ) ) ).

% rev_is_rev_conv
tff(fact_6837_rev__rev__ident,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),rev(A),Xs)) = Xs ).

% rev_rev_ident
tff(fact_6838_Nil__is__rev__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),rev(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% Nil_is_rev_conv
tff(fact_6839_rev__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rev_is_Nil_conv
tff(fact_6840_set__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rev
tff(fact_6841_length__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rev
tff(fact_6842_rev__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),rev(A),Xs)) ).

% rev_append
tff(fact_6843_distinct__rev,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),rev(A),Xs))
    <=> distinct(A,Xs) ) ).

% distinct_rev
tff(fact_6844_rev__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),list(A),rev(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ).

% rev_replicate
tff(fact_6845_remdups__adj__rev,axiom,
    ! [A: $tType,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),remdups_adj(A,Xs)) ).

% remdups_adj_rev
tff(fact_6846_inj__on__rev,axiom,
    ! [A: $tType,A3: set(list(A))] : inj_on(list(A),list(A),rev(A),A3) ).

% inj_on_rev
tff(fact_6847_rev__singleton__conv,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
    <=> ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ).

% rev_singleton_conv
tff(fact_6848_singleton__rev__conv,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) = aa(list(A),list(A),rev(A),Xs) )
    <=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) = Xs ) ) ).

% singleton_rev_conv
tff(fact_6849_rev__eq__Cons__iff,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))) ) ) ).

% rev_eq_Cons_iff
tff(fact_6850_sorted__wrt__rev,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),rev(A),Xs))
    <=> sorted_wrt(A,aTP_Lamp_aft(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P),Xs) ) ).

% sorted_wrt_rev
tff(fact_6851_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_6852_sorted__upto,axiom,
    ! [M: int,N: int] : sorted_wrt(int,ord_less_eq(int),upto(M,N)) ).

% sorted_upto
tff(fact_6853_rev_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rev(A),nil(A)) = nil(A) ).

% rev.simps(1)
tff(fact_6854_rev__swap,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = Ys )
    <=> ( Xs = aa(list(A),list(A),rev(A),Ys) ) ) ).

% rev_swap
tff(fact_6855_zip__rev,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( zip(A,B,aa(list(A),list(A),rev(A),Xs),aa(list(B),list(B),rev(B),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),rev(product_prod(A,B)),zip(A,B,Xs,Ys)) ) ) ).

% zip_rev
tff(fact_6856_rev__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),rev(A),Xs)) ).

% rev_filter
tff(fact_6857_rev__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rev(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),rev(B),Xs)) ).

% rev_map
tff(fact_6858_rev__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),list(A),rev(A),aa(list(list(A)),list(A),concat(A),Xs)) = aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),rev(A)),aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) ).

% rev_concat
tff(fact_6859_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xs: list(B),A2: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),Xs),A2) = aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),aTP_Lamp_afu(fun(B,fun(A,A)),fun(A,fun(B,A)),F2)),A2),aa(list(B),list(B),rev(B),Xs)) ).

% foldr_conv_foldl
tff(fact_6860_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,A)),A2: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),A2),Xs) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),aTP_Lamp_afv(fun(A,fun(B,A)),fun(B,fun(A,A)),F2)),aa(list(B),list(B),rev(B),Xs)),A2) ).

% foldl_conv_foldr
tff(fact_6861_rev_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% rev.simps(2)
tff(fact_6862_take__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)),Xs)) ).

% take_rev
tff(fact_6863_rev__take,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),I),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)),aa(list(A),list(A),rev(A),Xs)) ).

% rev_take
tff(fact_6864_rev__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),I),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)),aa(list(A),list(A),rev(A),Xs)) ).

% rev_drop
tff(fact_6865_drop__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)),Xs)) ).

% drop_rev
tff(fact_6866_rev__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rev(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,N))) ) ) ).

% rev_nth
tff(fact_6867_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),rev(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),Y)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),rev(A),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat))),Y) ) ) ).

% rev_update
tff(fact_6868_sorted__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_6869_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))),aa(nat,A,nth(A,Xs),I4))) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_6870_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I))) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_6871_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I4))) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_6872_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( ( ( Xs = nil(A) )
             => ( aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),foldr(A,A),ord_max(A)),Xs),Y) = Y ) )
            & ( ( Xs != nil(A) )
             => ( aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),foldr(A,A),ord_max(A)),Xs),Y) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y) ) ) ) ) ) ).

% foldr_max_sorted
tff(fact_6873_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
     => ( ( ( Xs = nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = zero_zero(nat) ) )
        & ( ( Xs != nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat))) ) ) ) ) ).

% length_transpose_sorted
tff(fact_6874_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),filter2(list(A)),aTP_Lamp_aef(nat,fun(list(A),bool),I)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).

% transpose_column_length
tff(fact_6875_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ~ ! [L3: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L3))
               => ( ( aa(list(B),set(B),set2(B),L3) = A3 )
                 => ( aa(list(B),nat,size_size(list(B)),L3) != aa(set(B),nat,finite_card(B),A3) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_6876_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),aa(fun(list(A),nat),fun(list(list(A)),list(nat)),map(list(A),nat),size_size(list(A))),Xs)))
     => ( transpose(A,transpose(A,Xs)) = aa(list(list(A)),list(list(A)),aa(fun(list(A),bool),fun(list(list(A)),list(list(A))),takeWhile(list(A)),aTP_Lamp_aey(list(A),bool)),Xs) ) ) ).

% transpose_transpose
tff(fact_6877_takeWhile__idem,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) ).

% takeWhile_idem
tff(fact_6878_takeWhile__eq__all__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = Xs )
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) ) ) ).

% takeWhile_eq_all_conv
tff(fact_6879_takeWhile__append1,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) ) ) ) ).

% takeWhile_append1
tff(fact_6880_takeWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) )
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Ys)) ) ) ).

% takeWhile_append2
tff(fact_6881_takeWhile__replicate,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,N: nat] :
      ( ( pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = nil(A) ) ) ) ).

% takeWhile_replicate
tff(fact_6882_length__concat__rev,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),nat,size_size(list(A)),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) = aa(list(A),nat,size_size(list(A)),aa(list(list(A)),list(A),concat(A),Xs)) ).

% length_concat_rev
tff(fact_6883_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)) ) ) ).

% sorted_takeWhile
tff(fact_6884_takeWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),takeWhile(B),aa(fun(B,A),fun(B,bool),comp(A,bool,B,P),F2)),Xs)) ).

% takeWhile_map
tff(fact_6885_zip__takeWhile__fst,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(fun(product_prod(A,B),bool),fun(list(product_prod(A,B)),list(product_prod(A,B))),takeWhile(product_prod(A,B)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),bool),comp(A,bool,product_prod(A,B),P),product_fst(A,B))),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_fst
tff(fact_6886_zip__takeWhile__snd,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),P: fun(B,bool),Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),takeWhile(B),P),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(fun(product_prod(A,B),bool),fun(list(product_prod(A,B)),list(product_prod(A,B))),takeWhile(product_prod(A,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),bool),comp(B,bool,product_prod(A,B),P),product_snd(A,B))),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_snd
tff(fact_6887_takeWhile__tail,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A),L: list(A)] :
      ( ~ pp(aa(A,bool,P,X))
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L))) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) ) ) ).

% takeWhile_tail
tff(fact_6888_folding__insort__key_Oinj__on,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => inj_on(B,A,F2,S2) ) ).

% folding_insort_key.inj_on
tff(fact_6889_takeWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),nil(A)) = nil(A) ).

% takeWhile.simps(1)
tff(fact_6890_takeWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) ) ) ).

% takeWhile.simps(2)
tff(fact_6891_distinct__takeWhile,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)) ) ).

% distinct_takeWhile
tff(fact_6892_length__takeWhile__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_takeWhile_le
tff(fact_6893_takeWhile__eq__take,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))),Xs) ).

% takeWhile_eq_take
tff(fact_6894_takeWhile__cong,axiom,
    ! [A: $tType,L: list(A),K: list(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ( L = K )
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),L)))
           => ( pp(aa(A,bool,P,X4))
            <=> pp(aa(A,bool,Q,X4)) ) )
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),L) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),Q),K) ) ) ) ).

% takeWhile_cong
tff(fact_6895_set__takeWhileD,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
        & pp(aa(A,bool,P,X)) ) ) ).

% set_takeWhileD
tff(fact_6896_takeWhile__eq__Nil__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ~ pp(aa(A,bool,P,aa(list(A),A,hd(A),Xs))) ) ) ).

% takeWhile_eq_Nil_iff
tff(fact_6897_folding__insort__key_Odistinct__if__distinct__map,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( distinct(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
       => distinct(B,Xs) ) ) ).

% folding_insort_key.distinct_if_distinct_map
tff(fact_6898_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs)))
     => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))))) ) ).

% nth_length_takeWhile
tff(fact_6899_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_6900_takeWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X4)) )
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Ys)) ) )
      & ( ~ ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P,X2)) )
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) ) ) ) ).

% takeWhile_append
tff(fact_6901_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I2))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_6902_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I2))) ) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
         => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) )
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_6903_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_kg(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_6904_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),T2: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)))
         => ( aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(A,fun(B,bool),aTP_Lamp_afw(fun(B,A),fun(A,fun(B,bool)),F2),T2)),Xs) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),takeWhile(B),aa(A,fun(B,bool),aTP_Lamp_afw(fun(B,A),fun(A,fun(B,bool)),F2),T2)),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_6905_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),A3: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),L))
              & ( aa(list(B),set(B),set2(B),L) = A3 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A3) ) )
          <=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_6906_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)),S2))
       => ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B))))) = remove1(B,X,aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_6907_linorder_Osorted__key__list__of__set_Ocong,axiom,
    ! [B: $tType,A: $tType,Less_eq: fun(A,fun(A,bool))] : sorted8670434370408473282of_set(A,B,Less_eq) = sorted8670434370408473282of_set(A,B,Less_eq) ).

% linorder.sorted_key_list_of_set.cong
tff(fact_6908_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),A3: set(B),B3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),S2))
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3) = aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),B3) )
           => ( pp(aa(set(B),bool,finite_finite2(B),A3))
             => ( pp(aa(set(B),bool,finite_finite2(B),B3))
               => ( A3 = B3 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_6909_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_6910_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(list(B),set(B),set2(B),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3)) = A3 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_6911_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( aa(list(B),nat,size_size(list(B)),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3)) = aa(set(B),nat,finite_card(B),A3) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_6912_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => distinct(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_6913_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_6914_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => sorted_wrt(A,Less,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_6915_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3) = nil(B) )
          <=> ( A3 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_6916_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S2))
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
         => ( distinct(B,Xs)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),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_6917_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)),S2))
       => ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_6918_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)),S2))
       => ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),A3)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_6919_linorder_Oinsort__key_Ocong,axiom,
    ! [B: $tType,A: $tType,Less_eq: fun(A,fun(A,bool))] : insort_key(A,B,Less_eq) = insort_key(A,B,Less_eq) ).

% linorder.insort_key.cong
tff(fact_6920_folding__insort__key_Oinsort__key__commute,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F2: fun(B,A),X: B,Y: B] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),S2))
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),S2))
         => ( aa(fun(list(B),list(B)),fun(list(B),list(B)),comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Y)),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X)) = aa(fun(list(B),list(B)),fun(list(B),list(B)),comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X)),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Y)) ) ) ) ) ).

% folding_insort_key.insort_key_commute
tff(fact_6921_extract__def,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : extract(A,P,Xs) = aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(option(product_prod(list(A),product_prod(A,list(A)))),fun(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),case_list(option(product_prod(list(A),product_prod(A,list(A)))),A),none(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_afx(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P)),Xs)) ).

% extract_def
tff(fact_6922_lenlex__append2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Us: list(A),Xs: list(A),Ys: list(A)] :
      ( irrefl(A,R)
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys))),lenlex(A,R)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lenlex(A,R))) ) ) ).

% lenlex_append2
tff(fact_6923_dropWhile__idem,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) ).

% dropWhile_idem
tff(fact_6924_dropWhile__eq__Nil__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) = nil(A) )
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) ) ) ).

% dropWhile_eq_Nil_conv
tff(fact_6925_dropWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) )
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Ys) ) ) ).

% dropWhile_append2
tff(fact_6926_dropWhile__append1,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)),Ys) ) ) ) ).

% dropWhile_append1
tff(fact_6927_dropWhile__replicate,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,N: nat] :
      ( ( pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = nil(A) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X) ) ) ) ).

% dropWhile_replicate
tff(fact_6928_takeWhile__dropWhile__id,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)) = Xs ).

% takeWhile_dropWhile_id
tff(fact_6929_lexord__same__pref__if__irrefl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( irrefl(A,R2)
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R2)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R2))) ) ) ).

% lexord_same_pref_if_irrefl
tff(fact_6930_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)) ) ) ).

% sorted_dropWhile
tff(fact_6931_remdups__adj__Cons_H,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),remdups_adj(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_bp(A,fun(A,bool),X)),Xs))) ).

% remdups_adj_Cons'
tff(fact_6932_hd__dropWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) != nil(A) )
     => ~ pp(aa(A,bool,P,aa(list(A),A,hd(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)))) ) ).

% hd_dropWhile
tff(fact_6933_dropWhile__eq__self__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) = Xs )
    <=> ( ( Xs = nil(A) )
        | ~ pp(aa(A,bool,P,aa(list(A),A,hd(A),Xs))) ) ) ).

% dropWhile_eq_self_iff
tff(fact_6934_set__dropWhileD,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% set_dropWhileD
tff(fact_6935_dropWhile__cong,axiom,
    ! [A: $tType,L: list(A),K: list(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ( L = K )
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),L)))
           => ( pp(aa(A,bool,P,X4))
            <=> pp(aa(A,bool,Q,X4)) ) )
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),L) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),Q),K) ) ) ) ).

% dropWhile_cong
tff(fact_6936_length__dropWhile__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_dropWhile_le
tff(fact_6937_distinct__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)) ) ).

% distinct_dropWhile
tff(fact_6938_irrefl__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
    <=> ! [A7: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),A7)),R2)) ) ).

% irrefl_def
tff(fact_6939_irreflI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A4: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4)),R))
     => irrefl(A,R) ) ).

% irreflI
tff(fact_6940_dropWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),nil(A)) = nil(A) ).

% dropWhile.simps(1)
tff(fact_6941_irrefl__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
     => irrefl(list(A),lex(A,R2)) ) ).

% irrefl_lex
tff(fact_6942_lexord__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
     => irrefl(list(A),lexord(A,R)) ) ).

% lexord_irrefl
tff(fact_6943_dropWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) ) ) ).

% dropWhile.simps(2)
tff(fact_6944_dropWhile__append3,axiom,
    ! [A: $tType,P: fun(A,bool),Y: A,Xs: list(A),Ys: list(A)] :
      ( ~ pp(aa(A,bool,P,Y))
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ).

% dropWhile_append3
tff(fact_6945_dropWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),dropWhile(B),aa(fun(B,A),fun(B,bool),comp(A,bool,B,P),F2)),Xs)) ).

% dropWhile_map
tff(fact_6946_dropWhile__eq__Cons__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Y: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) )
        & ~ pp(aa(A,bool,P,Y)) ) ) ).

% dropWhile_eq_Cons_conv
tff(fact_6947_takeWhile__eq__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))))
         => ~ pp(aa(A,bool,P,X4)) )
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ) ) ).

% takeWhile_eq_filter
tff(fact_6948_dropWhile__eq__drop,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs))),Xs) ).

% dropWhile_eq_drop
tff(fact_6949_dropWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X4)) )
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Ys) ) )
      & ( ~ ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P,X2)) )
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)),Ys) ) ) ) ).

% dropWhile_append
tff(fact_6950_lexl__not__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: list(A)] :
      ( irrefl(A,R2)
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),X)),lex(A,R2))) ) ).

% lexl_not_refl
tff(fact_6951_find__dropWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : find(A,P,Xs) = aa(list(A),option(A),aa(fun(A,fun(list(A),option(A))),fun(list(A),option(A)),aa(option(A),fun(fun(A,fun(list(A),option(A))),fun(list(A),option(A))),case_list(option(A),A),none(A)),aTP_Lamp_afy(A,fun(list(A),option(A)))),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P)),Xs)) ).

% find_dropWhile
tff(fact_6952_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),remdups_adj(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_bp(A,fun(A,bool),Y)),Ys))) ).

% remdups_adj_append_dropWhile
tff(fact_6953_tl__remdups__adj,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( ( Ys != nil(A) )
     => ( aa(list(A),list(A),tl(A),remdups_adj(A,Ys)) = remdups_adj(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_afz(list(A),fun(A,bool),Ys)),aa(list(A),list(A),tl(A),Ys))) ) ) ).

% tl_remdups_adj
tff(fact_6954_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs))))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(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)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),P),Xs)))) ) ) ).

% dropWhile_nth
tff(fact_6955_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aa(A,fun(A,bool),aTP_Lamp_aeo(A,fun(A,bool)),X)),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),aa(A,fun(A,bool),aTP_Lamp_aeo(A,fun(A,bool)),X)),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_6956_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),aa(A,fun(A,bool),aTP_Lamp_aeo(A,fun(A,bool)),X)),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aa(A,fun(A,bool),aTP_Lamp_aeo(A,fun(A,bool)),X)),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_6957_partition__filter__conv,axiom,
    ! [A: $tType,F2: fun(A,bool),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,F2),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),F2),Xs)),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),F2)),Xs)) ).

% partition_filter_conv
tff(fact_6958_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( ! [X4: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
               => ( Y != aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),X4),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_adn(A,fun(list(A),fun(A,fun(list(A),A))),X4),Xs2)),Xs2) ) )
           => ~ ( ( X = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_6959_partition__filter1,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),Xs) ).

% partition_filter1
tff(fact_6960_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = case_option(A,A,undefined(A),aTP_Lamp_ki(A,A),Option) ).

% option.the_def
tff(fact_6961_hd__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),A,hd(A),List) = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),undefined(A)),aTP_Lamp_aga(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_6962_partition_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)) ).

% partition.simps(1)
tff(fact_6963_partition__P,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No4) )
     => ( ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Yes)))
           => pp(aa(A,bool,P,X2)) )
        & ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),No4)))
           => ~ pp(aa(A,bool,P,X2)) ) ) ) ).

% partition_P
tff(fact_6964_partition_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(fun(list(A),fun(list(A),product_prod(list(A),list(A)))),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),product_case_prod(list(A),list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_agb(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P),X)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) ).

% partition.simps(2)
tff(fact_6965_partition__filter2,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P)),Xs) ).

% partition_filter2
tff(fact_6966_Func__empty,axiom,
    ! [B: $tType,A: $tType,B3: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B3) = 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_agc(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_6967_partition__set,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No4) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes)),aa(list(A),set(A),set2(A),No4)) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% partition_set
tff(fact_6968_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa2: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa2) = Y )
         => ( ! [X4: A] :
                ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) )
               => ( Y != X4 ) )
           => ( ! [X4: A,Y2: A,Zs2: list(A)] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)) )
                 => ( Y != if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X4)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)))),X4,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2))) ) )
             => ~ ( ( Xa2 = nil(A) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_6969_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( accp(list(A),min_list_rel(A),X)
           => ( ! [X4: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
                 => ( ( Y = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),X4),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_adn(A,fun(list(A),fun(A,fun(list(A),A))),X4),Xs2)),Xs2) )
                   => ~ accp(list(A),min_list_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)) ) )
             => ~ ( ( X = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_6970_lists__length__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] : aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_agd(set(A),fun(nat,fun(list(A),bool)),A3),N)) = 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_acw(list(A),fun(A,list(A))))),product_Sigma(list(A),A,aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_at(set(A),fun(nat,fun(list(A),bool)),A3),N)),aTP_Lamp_age(set(A),fun(list(A),set(A)),A3))) ).

% lists_length_Suc_eq
tff(fact_6971_SigmaI,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B2: B,B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(A,set(B),B3,A2)))
       => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),product_Sigma(A,B,A3,B3))) ) ) ).

% SigmaI
tff(fact_6972_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),product_Sigma(A,B,A3,B3)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
        & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(A,set(B),B3,A2))) ) ) ).

% mem_Sigma_iff
tff(fact_6973_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B3: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B3) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty1
tff(fact_6974_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A3: set(A)] : product_Sigma(A,B,A3,aTP_Lamp_agf(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_6975_Times__empty,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( ( product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( B3 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_6976_finite__SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B3,A4))) )
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,B3))) ) ) ).

% finite_SigmaI
tff(fact_6977_fst__image__times,axiom,
    ! [B: $tType,A: $tType,B3: set(B),A3: set(A)] :
      ( ( ( B3 = bot_bot(set(B)) )
       => ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))) = bot_bot(set(A)) ) )
      & ( ( B3 != bot_bot(set(B)) )
       => ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))) = A3 ) ) ) ).

% fst_image_times
tff(fact_6978_snd__image__times,axiom,
    ! [B: $tType,A: $tType,A3: set(B),B3: set(A)] :
      ( ( ( A3 = bot_bot(set(B)) )
       => ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A3,aTP_Lamp_lo(set(A),fun(B,set(A)),B3))) = bot_bot(set(A)) ) )
      & ( ( A3 != bot_bot(set(B)) )
       => ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A3,aTP_Lamp_lo(set(A),fun(B,set(A)),B3))) = B3 ) ) ) ).

% snd_image_times
tff(fact_6979_set__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),product(A,B,Xs,Ys)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xs),aTP_Lamp_agh(list(B),fun(A,set(B)),Ys)) ).

% set_product
tff(fact_6980_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A2: A,A3: set(A),B2: B,B3: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3),aa(set(B),fun(A,set(B)),aTP_Lamp_agi(B,fun(set(B),fun(A,set(B))),B2),B3)) = 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),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A3,aa(set(B),fun(A,set(B)),aTP_Lamp_agi(B,fun(set(B),fun(A,set(B))),B2),B3))),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3),aTP_Lamp_agg(set(B),fun(A,set(B)),B3)))) ).

% insert_Times_insert
tff(fact_6981_card__SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B3,X4))) )
       => ( aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A3,B3)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ma(fun(A,set(B)),fun(A,nat),B3)),A3) ) ) ) ).

% card_SigmaI
tff(fact_6982_SigmaE,axiom,
    ! [A: $tType,B: $tType,C2: product_prod(A,B),A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),C2),product_Sigma(A,B,A3,B3)))
     => ~ ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => ! [Y2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y2),aa(A,set(B),B3,X4)))
               => ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2) ) ) ) ) ).

% SigmaE
tff(fact_6983_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),product_Sigma(A,B,A3,B3)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ).

% SigmaD1
tff(fact_6984_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),product_Sigma(A,B,A3,B3)))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(A,set(B),B3,A2))) ) ).

% SigmaD2
tff(fact_6985_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),product_Sigma(A,B,A3,B3)))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
         => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(A,set(B),B3,A2))) ) ) ).

% SigmaE2
tff(fact_6986_finite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B3))
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3)))) ) ) ).

% finite_cartesian_product
tff(fact_6987_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C3: set(A),A3: set(B),B3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),C3))
     => ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A3,aTP_Lamp_lo(set(A),fun(B,set(A)),C3))),product_Sigma(B,A,B3,aTP_Lamp_lo(set(A),fun(B,set(A)),C3))))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3)) ) ) ).

% Times_subset_cancel2
tff(fact_6988_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C3: set(A),B3: fun(A,set(B)),D6: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C3))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B3,X4)),aa(A,set(B),D6,X4))) )
       => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,B3)),product_Sigma(A,B,C3,D6))) ) ) ).

% Sigma_mono
tff(fact_6989_Restr__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( 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,B3,aTP_Lamp_agj(set(A),fun(A,set(A)),B3)))),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3))) ) ) ).

% Restr_subset
tff(fact_6990_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I6: set(A),X6: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I6,X6) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),I6))
         => ( aa(A,set(B),X6,X3) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_6991_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),C3: set(A),D6: set(B)] :
      ( ( product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3)) = product_Sigma(A,B,C3,aTP_Lamp_agg(set(B),fun(A,set(B)),D6)) )
    <=> ( ( ( A3 = C3 )
          & ( B3 = D6 ) )
        | ( ( ( A3 = bot_bot(set(A)) )
            | ( B3 = bot_bot(set(B)) ) )
          & ( ( C3 = bot_bot(set(A)) )
            | ( D6 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_6992_swap__product,axiom,
    ! [A: $tType,B: $tType,A3: set(B),B3: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_jn(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A3,aTP_Lamp_lo(set(A),fun(B,set(A)),B3))) = product_Sigma(A,B,B3,aTP_Lamp_agg(set(B),fun(A,set(B)),A3)) ).

% swap_product
tff(fact_6993_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),C3: set(B),B3: set(A),D6: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B3,aTP_Lamp_agg(set(B),fun(A,set(B)),D6))))
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( C3 = bot_bot(set(B)) )
        | ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C3),D6)) ) ) ) ).

% times_subset_iff
tff(fact_6994_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F2: fun(C,A),G: fun(D,B),A3: set(C),B3: set(D)] : aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_acj(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F2),G))),product_Sigma(C,D,A3,aTP_Lamp_agk(set(D),fun(C,set(D)),B3))) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F2),A3),aa(set(D),fun(A,set(B)),aTP_Lamp_agl(fun(D,B),fun(set(D),fun(A,set(B))),G),B3)) ).

% image_paired_Times
tff(fact_6995_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3))))
       => ( ( A2 = B2 )
          | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_6996_finite__SigmaI2,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_agm(set(A),fun(fun(A,set(B)),fun(A,bool)),A3),B3))))
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B3,A4))) )
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,B3))) ) ) ).

% finite_SigmaI2
tff(fact_6997_finite__cartesian__productD1,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))))
     => ( ( B3 != bot_bot(set(B)) )
       => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% finite_cartesian_productD1
tff(fact_6998_finite__cartesian__productD2,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))))
     => ( ( A3 != bot_bot(set(A)) )
       => pp(aa(set(B),bool,finite_finite2(B),B3)) ) ) ).

% finite_cartesian_productD2
tff(fact_6999_finite__cartesian__product__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))))
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( B3 = bot_bot(set(B)) )
        | ( pp(aa(set(A),bool,finite_finite2(A),A3))
          & pp(aa(set(B),bool,finite_finite2(B),B3)) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_7000_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: 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,A3,B3)) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_agm(set(A),fun(fun(A,set(B)),fun(A,bool)),A3),B3)) ).

% fst_image_Sigma
tff(fact_7001_refl__onI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3))))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R2)) )
       => refl_on(A,A3,R2) ) ) ).

% refl_onI
tff(fact_7002_refl__on__def,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A3,R2)
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3))))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R2)) ) ) ) ).

% refl_on_def
tff(fact_7003_subset__fst__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),S2: set(product_prod(A,B)),Y: B] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))),S2))
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),B3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S2))) ) ) ).

% subset_fst_imageI
tff(fact_7004_subset__snd__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),S2: set(product_prod(A,B)),X: A] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))),S2))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S2))) ) ) ).

% subset_snd_imageI
tff(fact_7005_card__cartesian__product__singleton,axiom,
    ! [A: $tType,B: $tType,X: A,A3: 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),X),bot_bot(set(A))),aTP_Lamp_agg(set(B),fun(A,set(B)),A3))) = aa(set(B),nat,finite_card(B),A3) ).

% card_cartesian_product_singleton
tff(fact_7006_Sigma__interval__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [A3: set(B),V2: fun(B,A),W2: A] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),inf_inf(set(product_prod(B,A))),product_Sigma(B,A,A3,aTP_Lamp_agn(fun(B,A),fun(B,set(A)),V2))),product_Sigma(B,A,A3,aa(A,fun(B,set(A)),aTP_Lamp_ago(fun(B,A),fun(A,fun(B,set(A))),V2),W2))) = bot_bot(set(product_prod(B,A))) ) ).

% Sigma_interval_disjoint
tff(fact_7007_sum_OSigma,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: fun(B,set(C)),G: fun(B,fun(C,A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),B3,X4))) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_agp(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B3),G)),A3) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7311177749621191930dd_sum(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A3,B3)) ) ) ) ) ).

% sum.Sigma
tff(fact_7008_prod_OSigma,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B3: fun(B,set(C)),G: fun(B,fun(C,A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A3))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),B3,X4))) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_agq(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B3),G)),A3) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7121269368397514597t_prod(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A3,B3)) ) ) ) ) ).

% prod.Sigma
tff(fact_7009_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] : product_Sigma(A,B,A3,B3) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(A),set(set(product_prod(A,B))),image2(A,set(product_prod(A,B)),aTP_Lamp_ags(fun(A,set(B)),fun(A,set(product_prod(A,B))),B3)),A3)) ).

% Sigma_def
tff(fact_7010_product__fold,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B3))
       => ( product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_agu(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B3),bot_bot(set(product_prod(A,B))),A3) ) ) ) ).

% product_fold
tff(fact_7011_uniformly__continuous__on__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S,F2)
        <=> filterlim(product_prod(A,A),product_prod(B,B),aa(fun(A,fun(A,product_prod(B,B))),fun(product_prod(A,A),product_prod(B,B)),product_case_prod(A,A,product_prod(B,B)),aTP_Lamp_agv(fun(A,B),fun(A,fun(A,product_prod(B,B))),F2)),topolo7806501430040627800ormity(B),aa(filter(product_prod(A,A)),filter(product_prod(A,A)),aa(filter(product_prod(A,A)),fun(filter(product_prod(A,A)),filter(product_prod(A,A))),inf_inf(filter(product_prod(A,A))),topolo7806501430040627800ormity(A)),principal(product_prod(A,A),product_Sigma(A,A,S,aTP_Lamp_agw(set(A),fun(A,set(A)),S))))) ) ) ).

% uniformly_continuous_on_uniformity
tff(fact_7012_infinite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),B3))
       => ~ pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3)))) ) ) ).

% infinite_cartesian_product
tff(fact_7013_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B3: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),bNF_Gr(A,B,A3,F2)),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3)) ) ).

% Gr_incl
tff(fact_7014_GrD2,axiom,
    ! [A: $tType,B: $tType,X: A,Fx: B,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A3,F2)))
     => ( aa(A,B,F2,X) = Fx ) ) ).

% GrD2
tff(fact_7015_GrD1,axiom,
    ! [B: $tType,A: $tType,X: A,Fx: B,A3: set(A),F2: fun(A,B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A3,F2)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ).

% GrD1
tff(fact_7016_Gr__def,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] : bNF_Gr(A,B,A3,F2) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_agx(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),A3),F2)) ).

% Gr_def
tff(fact_7017_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_di(nat,fun(nat,fun(nat,bool)),M))) = product_Sigma(nat,nat,set_ord_atMost(nat,M),aTP_Lamp_agy(nat,fun(nat,set(nat)),M)) ).

% pairs_le_eq_Sigma
tff(fact_7018_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R2: set(product_prod(A,A)),As3: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R2,As3)
        <=> ! [I4: A,J3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I4),J3)),R2))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,As3,I4)),aa(A,B,As3,J3))) ) ) ) ).

% relChain_def
tff(fact_7019_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less(nat))) ).

% natLess_def
tff(fact_7020_rotate__drop__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Xs)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Xs)) ).

% rotate_drop_take
tff(fact_7021_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),A3: 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_wj(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G)),A3) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F2),A3),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_agz(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F2),G),A3)) ).

% image_split_eq_Sigma
tff(fact_7022_vimageI,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: B,B2: A,B3: set(A)] :
      ( ( aa(B,A,F2,A2) = B2 )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),B3))
       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),vimage(B,A,F2,B3))) ) ) ).

% vimageI
tff(fact_7023_vimage__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),B3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),vimage(A,B,F2,B3)))
    <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),B3)) ) ).

% vimage_eq
tff(fact_7024_vimage__ident,axiom,
    ! [A: $tType,Y5: set(A)] : vimage(A,A,aTP_Lamp_ki(A,A),Y5) = Y5 ).

% vimage_ident
tff(fact_7025_vimage__Collect__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),P: fun(B,bool)] : vimage(A,B,F2,aa(fun(B,bool),set(B),collect(B),P)) = aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_aha(fun(A,B),fun(fun(B,bool),fun(A,bool)),F2),P)) ).

% vimage_Collect_eq
tff(fact_7026_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_7027_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_7028_vimage__Int,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B3)) ).

% vimage_Int
tff(fact_7029_vimage__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B3)) ).

% vimage_Un
tff(fact_7030_rotate__is__Nil__conv,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rotate_is_Nil_conv
tff(fact_7031_set__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rotate
tff(fact_7032_length__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate
tff(fact_7033_distinct__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs))
    <=> distinct(A,Xs) ) ).

% distinct_rotate
tff(fact_7034_vimage__const,axiom,
    ! [B: $tType,A: $tType,C2: B,A3: set(B)] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A3))
       => ( vimage(A,B,aTP_Lamp_ku(B,fun(A,B),C2),A3) = top_top(set(A)) ) )
      & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A3))
       => ( vimage(A,B,aTP_Lamp_ku(B,fun(A,B),C2),A3) = bot_bot(set(A)) ) ) ) ).

% vimage_const
tff(fact_7035_rotate__Suc,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,suc,N)),Xs) = aa(list(A),list(A),rotate1(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) ).

% rotate_Suc
tff(fact_7036_image__vimage__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(A)] : aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ).

% image_vimage_eq
tff(fact_7037_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_7038_rotate__id,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = Xs ) ) ).

% rotate_id
tff(fact_7039_vimage__if,axiom,
    ! [B: $tType,A: $tType,C2: B,A3: set(B),D2: B,B3: set(A)] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A3))
       => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A3))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ahb(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B3),A3) = top_top(set(A)) ) )
          & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A3))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ahb(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B3),A3) = B3 ) ) ) )
      & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),C2),A3))
       => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A3))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ahb(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B3),A3) = aa(set(A),set(A),uminus_uminus(set(A)),B3) ) )
          & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),D2),A3))
           => ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ahb(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B3),A3) = bot_bot(set(A)) ) ) ) ) ) ).

% vimage_if
tff(fact_7040_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,X: B,A3: set(B),F2: fun(B,set(A))] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
       => ( vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),product_Sigma(B,A,A3,F2)) = aa(B,set(A),F2,X) ) )
      & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
       => ( vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),product_Sigma(B,A,A3,F2)) = bot_bot(set(A)) ) ) ) ).

% Pair_vimage_Sigma
tff(fact_7041_vimage__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A2: B,B3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),B3)) = 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),A2),bot_bot(set(B))))),vimage(A,B,F2,B3)) ).

% vimage_insert
tff(fact_7042_vimage__singleton__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),B2: B] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B2),bot_bot(set(B))))))
    <=> ( aa(A,B,F2,A2) = B2 ) ) ).

% vimage_singleton_eq
tff(fact_7043_rotate__map,axiom,
    ! [A: $tType,B: $tType,N: nat,F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),rotate(B),N),Xs)) ).

% rotate_map
tff(fact_7044_rotate__append,axiom,
    ! [A: $tType,L: list(A),Q4: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(list(A),nat,size_size(list(A)),L)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),Q4)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Q4),L) ).

% rotate_append
tff(fact_7045_subset__vimage__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B3: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),vimage(A,B,F2,B3)))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,X3)),B3)) ) ) ).

% subset_vimage_iff
tff(fact_7046_vimage__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F2: fun(B,A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),vimage(B,A,F2,A3)),vimage(B,A,F2,B3))) ) ).

% vimage_mono
tff(fact_7047_rotate__rotate,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),M),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),Xs) ).

% rotate_rotate
tff(fact_7048_rotate__def,axiom,
    ! [A: $tType,N: nat] : aa(nat,fun(list(A),list(A)),rotate(A),N) = aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(nat,fun(fun(list(A),list(A)),fun(list(A),list(A))),compow(fun(list(A),list(A))),N),rotate1(A)) ).

% rotate_def
tff(fact_7049_vimage__Diff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B3)) ).

% vimage_Diff
tff(fact_7050_vimageD,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),vimage(A,B,F2,A3)))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),A3)) ) ).

% vimageD
tff(fact_7051_vimageE,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),B3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),vimage(A,B,F2,B3)))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,A2)),B3)) ) ).

% vimageE
tff(fact_7052_vimageI2,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: B,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,A2)),A3))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),vimage(B,A,F2,A3))) ) ).

% vimageI2
tff(fact_7053_vimage__Collect,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F2: fun(A,B),Q: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(B,bool,P,aa(A,B,F2,X4)))
        <=> pp(aa(A,bool,Q,X4)) )
     => ( vimage(A,B,F2,aa(fun(B,bool),set(B),collect(B),P)) = aa(fun(A,bool),set(A),collect(A),Q) ) ) ).

% vimage_Collect
tff(fact_7054_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),uminus_uminus(set(B)),A3)) = aa(set(A),set(A),uminus_uminus(set(A)),vimage(A,B,F2,A3)) ).

% vimage_Compl
tff(fact_7055_vimage__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),B3: set(B)] : vimage(A,B,F2,B3) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_ahc(fun(A,B),fun(set(B),fun(A,bool)),F2),B3)) ).

% vimage_def
tff(fact_7056_rotate1__rotate__swap,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate1(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),aa(list(A),list(A),rotate1(A),Xs)) ).

% rotate1_rotate_swap
tff(fact_7057_finite__vimage__Suc__iff,axiom,
    ! [F4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),vimage(nat,nat,suc,F4)))
    <=> pp(aa(set(nat),bool,finite_finite2(nat),F4)) ) ).

% finite_vimage_Suc_iff
tff(fact_7058_finite__vimageI,axiom,
    ! [B: $tType,A: $tType,F4: set(A),H: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( inj_on(B,A,H,top_top(set(B)))
       => pp(aa(set(B),bool,finite_finite2(B),vimage(B,A,H,F4))) ) ) ).

% finite_vimageI
tff(fact_7059_continuous__imp__open__vimage,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),B3: set(B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo1002775350975398744n_open(A,S)
           => ( topolo1002775350975398744n_open(B,B3)
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),vimage(A,B,F2,B3)),S))
               => topolo1002775350975398744n_open(A,vimage(A,B,F2,B3)) ) ) ) ) ) ).

% continuous_imp_open_vimage
tff(fact_7060_rotate__conv__mod,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% rotate_conv_mod
tff(fact_7061_vimage__Suc__insert__0,axiom,
    ! [A3: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),A3)) = vimage(nat,nat,suc,A3) ).

% vimage_Suc_insert_0
tff(fact_7062_vimage__inter__cong,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,B),G: fun(A,B),Y: set(B)] :
      ( ! [W: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),W),S2))
         => ( 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,Y)),S2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,G,Y)),S2) ) ) ).

% vimage_inter_cong
tff(fact_7063_finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( inj_on(B,A,H,A3)
       => pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H,F4)),A3))) ) ) ).

% finite_vimage_IntI
tff(fact_7064_image__subset__iff__subset__vimage,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B3))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),vimage(B,A,F2,B3))) ) ).

% image_subset_iff_subset_vimage
tff(fact_7065_image__vimage__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,A3))),A3)) ).

% image_vimage_subset
tff(fact_7066_vimage__subsetD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),B3: set(A),A3: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),vimage(B,A,F2,B3)),A3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3))) ) ) ).

% vimage_subsetD
tff(fact_7067_finite__vimageD,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F4: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),vimage(A,B,H,F4)))
     => ( ( aa(set(A),set(B),image2(A,B,H),top_top(set(A))) = top_top(set(B)) )
       => pp(aa(set(B),bool,finite_finite2(B),F4)) ) ) ).

% finite_vimageD
tff(fact_7068_surj__vimage__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => ( ( vimage(B,A,F2,A3) = bot_bot(set(B)) )
      <=> ( A3 = bot_bot(set(A)) ) ) ) ).

% surj_vimage_empty
tff(fact_7069_rotate__add,axiom,
    ! [A: $tType,M: nat,N: nat] : aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),M)),aa(nat,fun(list(A),list(A)),rotate(A),N)) ).

% rotate_add
tff(fact_7070_finite__vimageD_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),vimage(A,B,F2,A3)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),aa(set(A),set(B),image2(A,B,F2),top_top(set(A)))))
       => pp(aa(set(B),bool,finite_finite2(B),A3)) ) ) ).

% finite_vimageD'
tff(fact_7071_inf__img__fin__domE,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3)))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),A3))
       => ~ ! [Y2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),aa(set(B),set(A),image2(B,A,F2),A3)))
             => pp(aa(set(B),bool,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A)))))) ) ) ) ).

% inf_img_fin_domE
tff(fact_7072_inf__img__fin__dom,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3)))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),A3))
       => ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(set(B),set(A),image2(B,A,F2),A3)))
            & ~ pp(aa(set(B),bool,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A)))))) ) ) ) ).

% inf_img_fin_dom
tff(fact_7073_vimage__subsetI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(B),A3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F2),A3)))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),vimage(A,B,F2,B3)),A3)) ) ) ).

% vimage_subsetI
tff(fact_7074_finite__finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( ! [Y2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),F4))
           => pp(aa(set(B),bool,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),Y2),bot_bot(set(A))))),A3))) )
       => pp(aa(set(B),bool,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H,F4)),A3))) ) ) ).

% finite_finite_vimage_IntI
tff(fact_7075_vimage__eq__UN,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),B3: set(B)] : vimage(A,B,F2,B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ahd(fun(A,B),fun(B,set(A)),F2)),B3)) ).

% vimage_eq_UN
tff(fact_7076_inf__img__fin__domE_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3)))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),A3))
       => ~ ! [Y2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),aa(set(B),set(A),image2(B,A,F2),A3)))
             => pp(aa(set(B),bool,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),Y2),bot_bot(set(A))))),A3))) ) ) ) ).

% inf_img_fin_domE'
tff(fact_7077_inf__img__fin__dom_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3)))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),A3))
       => ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(set(B),set(A),image2(B,A,F2),A3)))
            & ~ pp(aa(set(B),bool,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),X4),bot_bot(set(A))))),A3))) ) ) ) ).

% inf_img_fin_dom'
tff(fact_7078_card__vimage__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),aa(set(A),set(B),image2(A,B,F2),top_top(set(A)))))
       => ( aa(set(A),nat,finite_card(A),vimage(A,B,F2,A3)) = aa(set(B),nat,finite_card(B),A3) ) ) ) ).

% card_vimage_inj
tff(fact_7079_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),D6: set(A),A3: set(B)] :
      ( inj_on(A,B,F2,D6)
     => ( pp(aa(set(B),bool,finite_finite2(B),A3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,A3)),D6))),aa(set(B),nat,finite_card(B),A3))) ) ) ).

% card_vimage_inj_on_le
tff(fact_7080_rotate__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)))),Xs)) ).

% rotate_rev
tff(fact_7081_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: B] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,bool),aTP_Lamp_ahe(fun(A,B),fun(B,fun(A,bool)),F2),A2))),bot_bot(set(A))))) ) ).

% inj_vimage_singleton
tff(fact_7082_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),A2: B] :
      ( inj_on(A,B,F2,A3)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_ahf(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F2),A3),A2))),bot_bot(set(A))))) ) ).

% inj_on_vimage_singleton
tff(fact_7083_inv__image__partition,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) )
     => ( ! [Y2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),aa(list(A),set(A),set2(A),Ys)))
           => ~ pp(aa(A,bool,P,Y2)) )
       => ( vimage(list(A),product_prod(list(A),list(A)),partition(A,P),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),insert2(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),bot_bot(set(product_prod(list(A),list(A)))))) = shuffles(A,Xs,Ys) ) ) ) ).

% inv_image_partition
tff(fact_7084_nth__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A),M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),M),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_7085_hd__rotate__conv__nth,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),rotate(A),N),Xs)) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% hd_rotate_conv_nth
tff(fact_7086_Restr__natLeq,axiom,
    ! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N)),aTP_Lamp_ahg(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ahh(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq
tff(fact_7087_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),arg(Z)),pi)) ) ).

% Arg_bounded
tff(fact_7088_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less_eq(nat))) ).

% natLeq_def
tff(fact_7089_Restr__natLeq2,axiom,
    ! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,N),aTP_Lamp_ahi(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ahh(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq2
tff(fact_7090_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(arg(Z)) )
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z)))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),arg(Z)),pi)) ) ) ).

% Arg_correct
tff(fact_7091_underS__I,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( ( I != J )
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),order_underS(A,R,J))) ) ) ).

% underS_I
tff(fact_7092_underS__E,axiom,
    ! [A: $tType,I: A,R: set(product_prod(A,A)),J: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),order_underS(A,R,J)))
     => ( ( I != J )
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R)) ) ) ).

% underS_E
tff(fact_7093_underS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : order_underS(A,R2,A2) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_ahj(set(product_prod(A,A)),fun(A,fun(A,bool)),R2),A2)) ).

% underS_def
tff(fact_7094_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,N) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N)) ).

% natLeq_underS_less
tff(fact_7095_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(X) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
         => ( arg(Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_7096_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => bij_betw(nat,complex,aTP_Lamp_ahk(nat,fun(nat,complex),N),set_ord_lessThan(nat,N),aa(fun(complex,bool),set(complex),collect(complex),aTP_Lamp_cs(nat,fun(complex,bool),N))) ) ).

% bij_betw_roots_unity
tff(fact_7097_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,root(N),real_V7770717601297561774m_norm(complex,C2)))),cis(divide_divide(real,arg(C2),aa(nat,real,semiring_1_of_nat(real),N))))),aa(fun(complex,bool),set(complex),collect(complex),aTP_Lamp_cs(nat,fun(complex,bool),N)),aa(fun(complex,bool),set(complex),collect(complex),aa(nat,fun(complex,bool),aTP_Lamp_il(complex,fun(nat,fun(complex,bool)),C2),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_7098_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F9: fun(B,A),F2: fun(A,B),A11: set(B)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => ( aa(B,A,F9,aa(A,B,F2,X4)) = X4 ) )
     => ( ! [X4: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A11))
           => ( aa(A,B,F2,aa(B,A,F9,X4)) = X4 ) )
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),A11))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F9),A11)),A3))
           => bij_betw(A,B,F2,A3,A11) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_7099_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),A11: set(B),B3: set(A),B13: set(B)] :
      ( bij_betw(A,B,F2,A3,A11)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => ( ( aa(set(A),set(B),image2(A,B,F2),B3) = B13 )
         => bij_betw(A,B,F2,B3,B13) ) ) ) ).

% bij_betw_subset
tff(fact_7100_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F9: fun(A,B),A11: set(A),A16: set(B),F2: fun(C,A),A3: set(C)] :
      ( bij_betw(A,B,F9,A11,A16)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F2),A3)),A11))
       => ( bij_betw(C,A,F2,A3,A11)
        <=> bij_betw(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,F9),F2),A3,A16) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_7101_bij__betw__same__card,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B)] :
      ( bij_betw(A,B,F2,A3,B3)
     => ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) ) ) ).

% bij_betw_same_card
tff(fact_7102_bij__betw__iff__card,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B3))
       => ( ? [F7: fun(A,B)] : bij_betw(A,B,F7,A3,B3)
        <=> ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) ) ) ) ) ).

% bij_betw_iff_card
tff(fact_7103_finite__same__card__bij,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),B3))
       => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) )
         => ? [H5: fun(A,B)] : bij_betw(A,B,H5,A3,B3) ) ) ) ).

% finite_same_card_bij
tff(fact_7104_bij__betw__finite,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B)] :
      ( bij_betw(A,B,F2,A3,B3)
     => ( pp(aa(set(A),bool,finite_finite2(A),A3))
      <=> pp(aa(set(B),bool,finite_finite2(B),B3)) ) ) ).

% bij_betw_finite
tff(fact_7105_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( bij_betw(A,B,F2,bot_bot(set(A)),A3)
     => ( A3 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_7106_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( bij_betw(A,B,F2,A3,bot_bot(set(B)))
     => ( A3 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_7107_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B2: A,A3: set(A),F2: fun(A,B),A11: set(B)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
     => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,B2)),A11))
       => ( bij_betw(A,B,F2,A3,A11)
        <=> bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A11),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_7108_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B2: A,A3: set(A),F2: fun(A,B),A11: set(B)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
     => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F2,B2)),A11))
       => ( bij_betw(A,B,F2,A3,A11)
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A11),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_7109_bij__betw__funpow,axiom,
    ! [A: $tType,F2: fun(A,A),S2: set(A),N: nat] :
      ( bij_betw(A,A,F2,S2,S2)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),S2,S2) ) ).

% bij_betw_funpow
tff(fact_7110_bij__fn,axiom,
    ! [A: $tType,F2: fun(A,A),N: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F2),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_7111_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C3: set(A),B3: set(B),D6: set(B)] :
      ( bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D6))
     => ( bij_betw(A,B,F2,C3,D6)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C3) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D6) = bot_bot(set(B)) )
           => bij_betw(A,B,F2,A3,B3) ) ) ) ) ).

% bij_betw_partition
tff(fact_7112_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B),C3: set(A),D6: set(B)] :
      ( bij_betw(A,B,F2,A3,B3)
     => ( bij_betw(A,B,F2,C3,D6)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D6) = 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)),A3),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D6)) ) ) ) ).

% bij_betw_combine
tff(fact_7113_finite__vimage__iff,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F4: set(B)] :
      ( bij_betw(A,B,H,top_top(set(A)),top_top(set(B)))
     => ( pp(aa(set(A),bool,finite_finite2(A),vimage(A,B,H,F4)))
      <=> pp(aa(set(B),bool,finite_finite2(B),F4)) ) ) ).

% finite_vimage_iff
tff(fact_7114_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C3: set(B),G: fun(A,B),B3: set(A),D6: set(B)] :
      ( bij_betw(A,B,F2,A3,C3)
     => ( bij_betw(A,B,G,B3,D6)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C3),D6) = 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_yv(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C3),D6)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_7115_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I6: set(A),A3: fun(A,set(B)),F2: fun(B,C),A11: fun(A,set(C))] :
      ( ! [I2: A,J2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I6))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,I2)),aa(A,set(B),A3,J2)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I2))) ) ) )
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I6))
           => bij_betw(B,C,F2,aa(A,set(B),A3,I2),aa(A,set(C),A11,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),A3),I6)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A11),I6))) ) ) ).

% bij_betw_UNION_chain
tff(fact_7116_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [H5: fun(A,A)] : bij_betw(A,A,H5,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_7117_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [H5: fun(A,A)] : bij_betw(A,A,H5,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_7118_vimage__subset__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(B),A3: set(A)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),vimage(A,B,F2,B3)),A3))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ).

% vimage_subset_eq
tff(fact_7119_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),M4))
     => ? [H5: fun(nat,A)] : bij_betw(nat,A,H5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M4)),M4) ) ).

% ex_bij_betw_nat_finite
tff(fact_7120_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),M4))
     => ? [H5: fun(nat,A)] : bij_betw(nat,A,H5,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M4)),M4) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_7121_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S5: set(B),T5: set(C),H: fun(B,C),S2: set(B),T3: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T5))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T3),T5))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S5))
                   => ( aa(C,A,G,aa(B,C,H,A4)) = zero_zero(A) ) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T5))
                     => ( aa(C,A,G,B4) = zero_zero(A) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_ahl(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T3) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_7122_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S5: set(B),T5: set(C),H: fun(B,C),S2: set(B),T3: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T5))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T3),T5))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),S5))
                   => ( aa(C,A,G,aa(B,C,H,A4)) = one_one(A) ) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T5))
                     => ( aa(C,A,G,B4) = one_one(A) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_ahm(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),T3) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_7123_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat),B3: set(A)] :
      ( distinct(A,Xs)
     => ( ( A3 = set_ord_lessThan(nat,aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B3 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A3,B3) ) ) ) ).

% bij_betw_nth
tff(fact_7124_ex__bij__betw__strict__mono__card,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M4: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),M4))
         => ~ ! [H5: fun(nat,A)] :
                ( bij_betw(nat,A,H5,set_ord_lessThan(nat,aa(set(A),nat,finite_card(A),M4)),M4)
               => ~ strict_mono_on(nat,A,H5,set_ord_lessThan(nat,aa(set(A),nat,finite_card(A),M4))) ) ) ) ).

% ex_bij_betw_strict_mono_card
tff(fact_7125_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F2,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3))
       => ( inj_on(B,A,G,B3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),B3)),A3))
           => ? [H5: fun(A,B)] : bij_betw(A,B,H5,A3,B3) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_7126_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero(complex) )
       => ( arg(Z) = zero_zero(real) ) )
      & ( ( Z != zero_zero(complex) )
       => ( arg(Z) = fChoice(real,aTP_Lamp_ahn(complex,fun(real,bool),Z)) ) ) ) ).

% Arg_def
tff(fact_7127_bij__betw__Suc,axiom,
    ! [M4: set(nat),N4: set(nat)] :
      ( bij_betw(nat,nat,suc,M4,N4)
    <=> ( aa(set(nat),set(nat),image2(nat,nat,suc),M4) = N4 ) ) ).

% bij_betw_Suc
tff(fact_7128_some__in__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),A3))),A3))
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_7129_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X6: fun(bool,A),Y5: fun(bool,A)] :
          ( pp(aa(fun(bool,A),bool,aa(fun(bool,A),fun(fun(bool,A),bool),ord_less_eq(fun(bool,A)),X6),Y5))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fFalse)),aa(bool,A,Y5,fFalse)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fTrue)),aa(bool,A,Y5,fTrue))) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_7130_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M4: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),M4))
     => ? [H5: fun(A,nat)] : bij_betw(A,nat,H5,M4,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M4))) ) ).

% ex_bij_betw_finite_nat
tff(fact_7131_arg__min__SOME__Min,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( lattic7623131987881927897min_on(A,B,F2,S2) = fChoice(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_aho(set(A),fun(fun(A,B),fun(A,bool)),S2),F2)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_7132_max__ext__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : max_ext(A,R) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_ahp(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R))) ).

% max_ext_eq
tff(fact_7133_prod_Oset__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(list(B),set(B),set2(B),Xs)) = groups5270119922927024881d_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),remdups(B,Xs))) ) ).

% prod.set_conv_list
tff(fact_7134_bex__empty,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ~ ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),bot_bot(set(A))))
          & pp(aa(A,bool,P,X2)) ) ).

% bex_empty
tff(fact_7135_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_gg(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% Eps_case_prod_eq
tff(fact_7136_finite__Collect__bex,axiom,
    ! [B: $tType,A: $tType,A3: set(A),Q: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_ahq(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A3),Q))))
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_xw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q),X3)))) ) ) ) ).

% finite_Collect_bex
tff(fact_7137_bex__UNIV,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),top_top(set(A))))
          & pp(aa(A,bool,P,X3)) )
    <=> ? [X_13: A] : pp(aa(A,bool,P,X_13)) ) ).

% bex_UNIV
tff(fact_7138_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_ahr(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),P))) ).

% split_paired_Eps
tff(fact_7139_image__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] : aa(set(A),set(B),image2(A,B,F2),A3) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_ahs(fun(A,B),fun(set(A),fun(B,bool)),F2),A3)) ).

% image_def
tff(fact_7140_vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] : vimage(A,B,F2,aa(set(A),set(B),image2(A,B,F2),A3)) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aht(fun(A,B),fun(set(A),fun(A,bool)),F2),A3)) ).

% vimage_image_eq
tff(fact_7141_prod__list__zero__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Xs: list(A)] :
          ( ( groups5270119922927024881d_list(A,Xs) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% prod_list_zero_iff
tff(fact_7142_Bex__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
            & pp(aa(A,bool,P,X3)) )
      <=> pp(finite_fold(A,bool,aTP_Lamp_ahu(fun(A,bool),fun(A,fun(bool,bool)),P),fFalse,A3)) ) ) ).

% Bex_fold
tff(fact_7143_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat),B3: set(nat)] : nths(A,nths(A,Xs,A3),B3) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ahw(set(nat),fun(set(nat),fun(nat,bool)),A3),B3))) ).

% nths_nths
tff(fact_7144_max__extp_Ocases,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),A1: set(A),A22: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R),A1),A22))
     => ~ ( pp(aa(set(A),bool,finite_finite2(A),A1))
         => ( pp(aa(set(A),bool,finite_finite2(A),A22))
           => ( ( A22 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
             => ~ ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A1))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A22))
                        & pp(aa(A,bool,aa(A,fun(A,bool),R,X2),Xa4)) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_7145_max__extp_Osimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),A1: set(A),A22: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R),A1),A22))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A1))
        & pp(aa(set(A),bool,finite_finite2(A),A22))
        & ( A22 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A1))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A22))
                & pp(aa(A,bool,aa(A,fun(A,bool),R,X3),Xa3)) ) ) ) ) ).

% max_extp.simps
tff(fact_7146_max__extp_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y5: set(A),R: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),Y5))
       => ( ( Y5 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => ? [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),Y5))
                    & pp(aa(A,bool,aa(A,fun(A,bool),R,X4),Xa)) ) )
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R),X6),Y5)) ) ) ) ) ).

% max_extp.max_extI
tff(fact_7147_prod_Odistinct__set__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Xs: list(B),G: fun(B,A)] :
          ( distinct(B,Xs)
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(list(B),set(B),set2(B),Xs)) = groups5270119922927024881d_list(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs)) ) ) ) ).

% prod.distinct_set_conv_list
tff(fact_7148_min__ext__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : min_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_ahx(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),R2)) ).

% min_ext_def
tff(fact_7149_map__project__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),A3: set(A)] : map_project(A,B,F2,A3) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_ahy(fun(A,option(B)),fun(set(A),fun(B,bool)),F2),A3)) ).

% map_project_def
tff(fact_7150_to__nat__on__def,axiom,
    ! [A: $tType,S2: set(A)] : countable_to_nat_on(A,S2) = fChoice(fun(A,nat),aTP_Lamp_ahz(set(A),fun(fun(A,nat),bool),S2)) ).

% to_nat_on_def
tff(fact_7151_to__nat__on__finite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => bij_betw(A,nat,countable_to_nat_on(A,S2),S2,set_ord_lessThan(nat,aa(set(A),nat,finite_card(A),S2))) ) ).

% to_nat_on_finite
tff(fact_7152_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the(product_prod(int,int),aTP_Lamp_aia(rat,fun(product_prod(int,int),bool),X)) ).

% quotient_of_def
tff(fact_7153_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_7154_Cons__in__lists__iff,axiom,
    ! [A: $tType,X: A,Xs: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),lists(A,A3)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
        & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3))) ) ) ).

% Cons_in_lists_iff
tff(fact_7155_in__listsI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3)) )
     => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3))) ) ).

% in_listsI
tff(fact_7156_lists__Int__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,A3)),lists(A,B3)) ).

% lists_Int_eq
tff(fact_7157_append__in__lists__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),lists(A,A3)))
    <=> ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3)))
        & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys),lists(A,A3))) ) ) ).

% append_in_lists_conv
tff(fact_7158_coprime__power__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,N: nat] :
          ( algebr8660921524188924756oprime(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
        <=> ( algebr8660921524188924756oprime(A,A2,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_7159_coprime__power__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,N: nat,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N),B2)
        <=> ( algebr8660921524188924756oprime(A,A2,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_7160_coprime__mod__right__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,A2,modulo_modulo(A,B2,A2))
          <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mod_right_iff
tff(fact_7161_coprime__mod__left__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,modulo_modulo(A,A2,B2),B2)
          <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mod_left_iff
tff(fact_7162_lists__UNIV,axiom,
    ! [A: $tType] : lists(A,top_top(set(A))) = top_top(set(list(A))) ).

% lists_UNIV
tff(fact_7163_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,A2,zero_zero(A))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A))) ) ) ).

% coprime_0_right_iff
tff(fact_7164_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A2)
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),one_one(A))) ) ) ).

% coprime_0_left_iff
tff(fact_7165_normalize__stable,axiom,
    ! [Q4: int,P2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q4))
     => ( algebr8660921524188924756oprime(int,P2,Q4)
       => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4) ) ) ) ).

% normalize_stable
tff(fact_7166_in__listsD,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3)))
     => ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3)) ) ) ).

% in_listsD
tff(fact_7167_in__lists__conv__set,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3)))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3)) ) ) ).

% in_lists_conv_set
tff(fact_7168_normalize__coprime,axiom,
    ! [R2: product_prod(int,int),P2: int,Q4: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4) )
     => algebr8660921524188924756oprime(int,P2,Q4) ) ).

% normalize_coprime
tff(fact_7169_quotient__of__coprime,axiom,
    ! [R2: rat,P2: int,Q4: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q4) )
     => algebr8660921524188924756oprime(int,P2,Q4) ) ).

% quotient_of_coprime
tff(fact_7170_lists_Ocases,axiom,
    ! [A: $tType,A2: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),A2),lists(A,A3)))
     => ( ( A2 != nil(A) )
       => ~ ! [A4: A,L3: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),L3) )
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
               => ~ pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L3),lists(A,A3))) ) ) ) ) ).

% lists.cases
tff(fact_7171_lists_Osimps,axiom,
    ! [A: $tType,A2: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),A2),lists(A,A3)))
    <=> ( ( A2 = nil(A) )
        | ? [A7: A,L4: list(A)] :
            ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),L4) )
            & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),A3))
            & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L4),lists(A,A3))) ) ) ) ).

% lists.simps
tff(fact_7172_lists_ONil,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),nil(A)),lists(A,A3))) ).

% lists.Nil
tff(fact_7173_listsE,axiom,
    ! [A: $tType,X: A,L: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L)),lists(A,A3)))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ~ pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,A3))) ) ) ).

% listsE
tff(fact_7174_lists_OCons,axiom,
    ! [A: $tType,A2: A,A3: set(A),L: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,A3)))
       => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),L)),lists(A,A3))) ) ) ).

% lists.Cons
tff(fact_7175_listrel__refl__on,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A3,R2)
     => refl_on(list(A),lists(A,A3),listrel(A,A,R2)) ) ).

% listrel_refl_on
tff(fact_7176_div__gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( ( A2 != zero_zero(A) )
            | ( B2 != zero_zero(A) ) )
         => algebr8660921524188924756oprime(A,divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),divide_divide(A,B2,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2))) ) ) ).

% div_gcd_coprime
tff(fact_7177_lists__IntI,axiom,
    ! [A: $tType,L: list(A),A3: set(A),B3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,A3)))
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,B3)))
       => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))) ) ) ).

% lists_IntI
tff(fact_7178_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,A6: A,B6: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),A6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
           => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
             => algebr8660921524188924756oprime(A,A6,B6) ) ) ) ) ).

% gcd_coprime
tff(fact_7179_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) != zero_zero(A) )
         => ? [A17: A,B14: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),A17),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
              & ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B14),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)) )
              & algebr8660921524188924756oprime(A,A17,B14) ) ) ) ).

% gcd_coprime_exists
tff(fact_7180_prod__list__coprime__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),A2: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
             => algebr8660921524188924756oprime(A,A2,X4) )
         => algebr8660921524188924756oprime(A,A2,groups5270119922927024881d_list(A,Xs)) ) ) ).

% prod_list_coprime_right
tff(fact_7181_prod__list__coprime__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),A2: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
             => algebr8660921524188924756oprime(A,X4,A2) )
         => algebr8660921524188924756oprime(A,groups5270119922927024881d_list(A,Xs),A2) ) ) ).

% prod_list_coprime_left
tff(fact_7182_lists__eq__set,axiom,
    ! [A: $tType,A3: set(A)] : lists(A,A3) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_aib(set(A),fun(list(A),bool),A3)) ).

% lists_eq_set
tff(fact_7183_Rat__induct,axiom,
    ! [P: fun(rat,bool),Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
         => ( algebr8660921524188924756oprime(int,A4,B4)
           => pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A4),B4))) ) )
     => pp(aa(rat,bool,P,Q4)) ) ).

% Rat_induct
tff(fact_7184_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A4: int,B4: int] :
          ( ( Q4 = aa(int,rat,aa(int,fun(int,rat),fract,A4),B4) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
           => ~ algebr8660921524188924756oprime(int,A4,B4) ) ) ).

% Rat_cases
tff(fact_7185_lists__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),lists(A,A3)),lists(A,B3))) ) ).

% lists_mono
tff(fact_7186_lists__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : lists(A,aa(set(B),set(A),image2(B,A,F2),A3)) = aa(set(list(B)),set(list(A)),image2(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2)),lists(B,A3)) ).

% lists_image
tff(fact_7187_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( ( Q4 = aa(int,rat,aa(int,fun(int,rat),fract,A4),B4) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
           => ( ( A4 != zero_zero(int) )
             => ~ algebr8660921524188924756oprime(int,A4,B4) ) ) )
     => ( Q4 = zero_zero(rat) ) ) ).

% Rat_cases_nonzero
tff(fact_7188_Rats__cases_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),field_char_0_Rats(A)))
         => ~ ! [A4: int,B4: int] :
                ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
               => ( algebr8660921524188924756oprime(int,A4,B4)
                 => ( X != divide_divide(A,aa(int,A,ring_1_of_int(A),A4),aa(int,A,ring_1_of_int(A),B4)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_7189_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X4: product_prod(int,int)] :
      ( ( R2 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),X4)),aa(product_prod(int,int),int,product_snd(int,int),X4)) )
      & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),X4)))
      & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),X4),aa(product_prod(int,int),int,product_snd(int,int),X4))
      & ! [Y3: product_prod(int,int)] :
          ( ( ( R2 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Y3)),aa(product_prod(int,int),int,product_snd(int,int),Y3)) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Y3)))
            & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y3),aa(product_prod(int,int),int,product_snd(int,int),Y3)) )
         => ( Y3 = X4 ) ) ) ).

% quotient_of_unique
tff(fact_7190_Collect__finite__eq__lists,axiom,
    ! [A: $tType] : aa(fun(set(A),bool),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_7191_Collect__finite__subset__eq__lists,axiom,
    ! [A: $tType,T3: set(A)] : aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_za(set(A),fun(set(A),bool),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_7192_listrel__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3))))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R2)),product_Sigma(list(A),list(A),lists(A,A3),aTP_Lamp_aic(set(A),fun(list(A),set(list(A))),A3)))) ) ).

% listrel_subset
tff(fact_7193_card__quotient__disjoint,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( inj_on(A,set(set(A)),aTP_Lamp_aid(set(product_prod(A,A)),fun(A,set(set(A))),R2),A3)
       => ( aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A3,R2)) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_quotient_disjoint
tff(fact_7194_INF__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F2)),Xs),top_top(A)) ) ).

% INF_set_fold
tff(fact_7195_fold__append,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xs: list(B),Ys: list(B)] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Ys)),aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Xs)) ).

% fold_append
tff(fact_7196_fold__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),N: nat,X: B] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(B,list(B),aa(nat,fun(B,list(B)),replicate(B),N),X)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(B,fun(A,A),F2,X)) ).

% fold_replicate
tff(fact_7197_coprime__Suc__0__right,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_7198_coprime__Suc__0__left,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),N) ).

% coprime_Suc_0_left
tff(fact_7199_fold__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(B,fun(A,A)),F2: fun(C,B),Xs: list(C)] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),G),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F2),Xs)) = aa(list(C),fun(A,A),aa(fun(C,fun(A,A)),fun(list(C),fun(A,A)),fold(C,A),aa(fun(C,B),fun(C,fun(A,A)),comp(B,fun(A,A),C,G),F2)),Xs) ).

% fold_map
tff(fact_7200_fold__commute__apply,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F2: fun(A,fun(C,C)),S: B] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( aa(fun(B,B),fun(B,C),comp(B,C,B,H),aa(A,fun(B,B),G,X4)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F2,X4)),H) ) )
     => ( aa(B,C,H,aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),G),Xs),S)) = aa(C,C,aa(list(A),fun(C,C),aa(fun(A,fun(C,C)),fun(list(A),fun(C,C)),fold(A,C),F2),Xs),aa(B,C,H,S)) ) ) ).

% fold_commute_apply
tff(fact_7201_fold__commute,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F2: fun(A,fun(C,C))] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( aa(fun(B,B),fun(B,C),comp(B,C,B,H),aa(A,fun(B,B),G,X4)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F2,X4)),H) ) )
     => ( aa(fun(B,B),fun(B,C),comp(B,C,B,H),aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),G),Xs)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(list(A),fun(C,C),aa(fun(A,fun(C,C)),fun(list(A),fun(C,C)),fold(A,C),F2),Xs)),H) ) ) ).

% fold_commute
tff(fact_7202_fold__Cons,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),X: A,Xs: list(A)] : aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xs)),aa(A,fun(B,B),F2,X)) ).

% fold_Cons
tff(fact_7203_fold__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),X: B,Xs: list(B),S: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),S) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Xs),aa(A,A,aa(B,fun(A,A),F2,X),S)) ).

% fold_simps(2)
tff(fact_7204_foldl__conv__fold,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,A)),S: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(A,fun(B,A)),fun(A,fun(list(B),A)),foldl(A,B),F2),S),Xs) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),aTP_Lamp_afv(fun(A,fun(B,A)),fun(B,fun(A,A)),F2)),Xs),S) ).

% foldl_conv_fold
tff(fact_7205_fold__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),S: A] : aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),nil(B)),S) = S ).

% fold_simps(1)
tff(fact_7206_fold__invariant,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Q: fun(A,bool),P: fun(B,bool),S: B,F2: fun(A,fun(B,B))] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,Q,X4)) )
     => ( pp(aa(B,bool,P,S))
       => ( ! [X4: A,S4: B] :
              ( pp(aa(A,bool,Q,X4))
             => ( pp(aa(B,bool,P,S4))
               => pp(aa(B,bool,P,aa(B,B,aa(A,fun(B,B),F2,X4),S4))) ) )
         => pp(aa(B,bool,P,aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xs),S))) ) ) ) ).

% fold_invariant
tff(fact_7207_List_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,Xs: list(B),Ys: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( Xs = Ys )
       => ( ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(list(B),set(B),set2(B),Xs)))
             => ( aa(B,fun(A,A),F2,X4) = aa(B,fun(A,A),G,X4) ) )
         => ( aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),Xs),A2) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),G),Ys),B2) ) ) ) ) ).

% List.fold_cong
tff(fact_7208_foldr__conv__fold,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xs: list(B)] : aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),foldr(B,A),F2),Xs) = aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),fold(B,A),F2),aa(list(B),list(B),rev(B),Xs)) ).

% foldr_conv_fold
tff(fact_7209_union__set__fold,axiom,
    ! [A: $tType,Xs: list(A),A3: 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)),A3) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),aa(fun(A,fun(set(A),set(A))),fun(list(A),fun(set(A),set(A))),fold(A,set(A)),insert2(A)),Xs),A3) ).

% union_set_fold
tff(fact_7210_rev__conv__fold,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),cons(A)),Xs),nil(A)) ).

% rev_conv_fold
tff(fact_7211_fold__Cons__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),cons(A)),Xs) = aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xs)) ).

% fold_Cons_rev
tff(fact_7212_fold__rev,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y2)),aa(A,fun(B,B),F2,X4)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X4)),aa(A,fun(B,B),F2,Y2)) ) ) )
     => ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xs) ) ) ).

% fold_rev
tff(fact_7213_fold__remove1__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,fun(B,B)),X: A] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X4)),aa(A,fun(B,B),F2,Y2)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y2)),aa(A,fun(B,B),F2,X4)) ) ) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xs) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),remove1(A,X,Xs))),aa(A,fun(B,B),F2,X)) ) ) ) ).

% fold_remove1_split
tff(fact_7214_foldr__fold,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y2)),aa(A,fun(B,B),F2,X4)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X4)),aa(A,fun(B,B),F2,Y2)) ) ) )
     => ( aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),foldr(A,B),F2),Xs) = aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xs) ) ) ).

% foldr_fold
tff(fact_7215_minus__set__fold,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(list(A),set(A),set2(A),Xs)) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),aa(fun(A,fun(set(A),set(A))),fun(list(A),fun(set(A),set(A))),fold(A,set(A)),remove(A)),Xs),A3) ).

% minus_set_fold
tff(fact_7216_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_7217_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),N) ) ).

% coprime_diff_one_left_nat
tff(fact_7218_fold__append__concat__rev,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),fun(list(A),list(A)),aa(fun(list(A),fun(list(A),list(A))),fun(list(list(A)),fun(list(A),list(A))),fold(list(A),list(A)),append(A)),Xss) = aa(list(A),fun(list(A),list(A)),append(A),aa(list(list(A)),list(A),concat(A),aa(list(list(A)),list(list(A)),rev(list(A)),Xss))) ).

% fold_append_concat_rev
tff(fact_7219_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)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),sup_sup(A)),Xs),bot_bot(A)) ) ).

% Sup_set_fold
tff(fact_7220_Inf__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),inf_inf(A)),Xs),top_top(A)) ) ).

% Inf_set_fold
tff(fact_7221_Gcd__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : gcd_Gcd(A,aa(list(A),set(A),set2(A),Xs)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),gcd_gcd(A)),Xs),zero_zero(A)) ) ).

% Gcd_set_eq_fold
tff(fact_7222_Inf__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),inf_inf(A)),Xs),X) ) ).

% Inf_fin.set_eq_fold
tff(fact_7223_Sup__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),sup_sup(A)),Xs),X) ) ).

% Sup_fin.set_eq_fold
tff(fact_7224_Max_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),ord_max(A)),Xs),X) ) ).

% Max.set_eq_fold
tff(fact_7225_Min_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),ord_min(A)),Xs),X) ) ).

% Min.set_eq_fold
tff(fact_7226_comp__fun__idem__on_Ofold__set__fold,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S2))
       => ( finite_fold(A,B,F2,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),Xs),Y) ) ) ) ).

% comp_fun_idem_on.fold_set_fold
tff(fact_7227_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Gcd(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,aa(list(nat),fun(nat,nat),aa(fun(nat,fun(nat,nat)),fun(list(nat),fun(nat,nat)),fold(nat,nat),gcd_gcd(nat)),Xs),zero_zero(nat)) ).

% Gcd_nat_set_eq_fold
tff(fact_7228_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),field_char_0_Rats(real)))
     => ~ ! [M5: nat,N2: nat] :
            ( ( N2 != zero_zero(nat) )
           => ( ( aa(real,real,abs_abs(real),X) = divide_divide(real,aa(nat,real,semiring_1_of_nat(real),M5),aa(nat,real,semiring_1_of_nat(real),N2)) )
             => ~ algebr8660921524188924756oprime(nat,M5,N2) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_7229_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S2))
       => ( finite_fold(A,B,F2,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(fun(A,fun(B,B)),fun(list(A),fun(B,B)),fold(A,B),F2),remdups(A,Xs)),Y) ) ) ) ).

% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_7230_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))) = aa(A,A,aa(list(B),fun(A,A),aa(fun(B,fun(A,A)),fun(list(B),fun(A,A)),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_7231_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_7232_quotient__is__empty,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( ( equiv_quotient(A,A3,R2) = bot_bot(set(set(A))) )
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% quotient_is_empty
tff(fact_7233_quotient__is__empty2,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( ( bot_bot(set(set(A))) = equiv_quotient(A,A3,R2) )
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% quotient_is_empty2
tff(fact_7234_finite__equiv__class,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3))))
       => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
         => pp(aa(set(A),bool,finite_finite2(A),X6)) ) ) ) ).

% finite_equiv_class
tff(fact_7235_finite__quotient,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3))))
       => pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A3,R2))) ) ) ).

% finite_quotient
tff(fact_7236_quotient__diff1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),A2: A] :
      ( inj_on(A,set(set(A)),aTP_Lamp_aid(set(product_prod(A,A)),fun(A,set(set(A))),R2),A3)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => ( equiv_quotient(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))),R2) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),minus_minus(set(set(A))),equiv_quotient(A,A3,R2)),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))),R2)) ) ) ) ).

% quotient_diff1
tff(fact_7237_sort__key__conv__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] :
          ( inj_on(B,A,F2,aa(list(B),set(B),set2(B),Xs))
         => ( aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),aa(fun(B,fun(list(B),list(B))),fun(list(B),fun(list(B),list(B))),fold(B,list(B)),linorder_insort_key(B,A,F2)),Xs),nil(B)) ) ) ) ).

% sort_key_conv_fold
tff(fact_7238_inter__coset__fold,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),coset(A,Xs)) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),aa(fun(A,fun(set(A),set(A))),fun(list(A),fun(set(A),set(A))),fold(A,set(A)),remove(A)),Xs),A3) ).

% inter_coset_fold
tff(fact_7239_sort__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),linorder_sort_key(nat,nat),aTP_Lamp_co(nat,nat)),upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_7240_sort__upto,axiom,
    ! [I: int,J: int] : aa(list(int),list(int),aa(fun(int,int),fun(list(int),list(int)),linorder_sort_key(int,int),aTP_Lamp_aie(int,int)),upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_7241_sort__key__simps_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A)] : aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),nil(B)) = nil(B) ) ).

% sort_key_simps(1)
tff(fact_7242_set__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] : aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs)) = aa(list(B),set(B),set2(B),Xs) ) ).

% set_sort
tff(fact_7243_length__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ) ).

% length_sort
tff(fact_7244_distinct__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] :
          ( distinct(B,aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs))
        <=> distinct(B,Xs) ) ) ).

% distinct_sort
tff(fact_7245_sort__key__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] : aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs)) ) ).

% sort_key_simps(2)
tff(fact_7246_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),K: B,Xs: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(B,fun(A,bool),aTP_Lamp_aif(fun(A,B),fun(B,fun(A,bool)),F2),K)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aa(B,fun(A,bool),aTP_Lamp_aif(fun(A,B),fun(B,fun(A,bool)),F2),K)),Xs) ) ).

% sort_key_stable
tff(fact_7247_filter__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),F2: fun(B,A),Xs: list(B)] : aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs)) = aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),P),Xs)) ) ).

% filter_sort
tff(fact_7248_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xs: list(A)] : aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),aTP_Lamp_aig(B,fun(A,B),C2)),Xs) = Xs ) ).

% sort_key_const
tff(fact_7249_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_kg(A,A)),Xs)) ) ).

% sorted_sort
tff(fact_7250_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_kg(A,A)),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_7251_UNIV__coset,axiom,
    ! [A: $tType] : top_top(set(A)) = coset(A,nil(A)) ).

% UNIV_coset
tff(fact_7252_subset__code_I2_J,axiom,
    ! [B: $tType,A3: set(B),Ys: list(B)] :
      ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),coset(B,Ys)))
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Ys)))
         => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3)) ) ) ).

% subset_code(2)
tff(fact_7253_compl__coset,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),set(A),uminus_uminus(set(A)),coset(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% compl_coset
tff(fact_7254_coset__def,axiom,
    ! [A: $tType,Xs: list(A)] : coset(A,Xs) = aa(set(A),set(A),uminus_uminus(set(A)),aa(list(A),set(A),set2(A),Xs)) ).

% coset_def
tff(fact_7255_insert__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),coset(A,Xs)) = coset(A,aa(list(A),list(A),removeAll(A,X),Xs)) ).

% insert_code(2)
tff(fact_7256_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs))) ) ).

% sorted_sort_key
tff(fact_7257_union__coset__filter,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),coset(A,Xs)),A3) = coset(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_bv(set(A),fun(A,bool),A3)),Xs)) ).

% union_coset_filter
tff(fact_7258_subset__code_I3_J,axiom,
    ! [C: $tType] : ~ pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),coset(C,nil(C))),aa(list(C),set(C),set2(C),nil(C)))) ).

% subset_code(3)
tff(fact_7259_sorted__list__of__set__sort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_kg(A,A)),remdups(A,Xs)) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_7260_sort__conv__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_kg(A,A)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A))),Xs),nil(A)) ) ).

% sort_conv_fold
tff(fact_7261_minus__coset__filter,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),coset(A,Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),aTP_Lamp_a(set(A),fun(A,bool),A3)),Xs)) ).

% minus_coset_filter
tff(fact_7262_sort__key__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] : aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),aa(fun(B,fun(list(B),list(B))),fun(list(B),fun(list(B),list(B))),foldr(B,list(B)),linorder_insort_key(B,A,F2)),Xs),nil(B)) ) ).

% sort_key_def
tff(fact_7263_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] : bleast(A,aa(list(A),set(A),set2(A),Xs),P) = aa(list(A),A,aa(fun(A,fun(list(A),A)),fun(list(A),A),aa(A,fun(fun(A,fun(list(A),A)),fun(list(A),A)),case_list(A,A),abort_Bleast(A,aa(list(A),set(A),set2(A),Xs),P)),aTP_Lamp_aih(A,fun(list(A),A))),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter2(A),P),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_kg(A,A)),Xs))) ) ).

% Bleast_code
tff(fact_7264_finite__sequence__to__countable__set,axiom,
    ! [A: $tType,X6: set(A)] :
      ( countable_countable(A,X6)
     => ~ ! [F5: fun(nat,set(A))] :
            ( ! [I3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F5,I3)),X6))
           => ( ! [I3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F5,I3)),aa(nat,set(A),F5,aa(nat,nat,suc,I3))))
             => ( ! [I3: nat] : pp(aa(set(A),bool,finite_finite2(A),aa(nat,set(A),F5,I3)))
               => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),F5),top_top(set(nat)))) != X6 ) ) ) ) ) ).

% finite_sequence_to_countable_set
tff(fact_7265_countable__empty,axiom,
    ! [A: $tType] : countable_countable(A,bot_bot(set(A))) ).

% countable_empty
tff(fact_7266_countable__Diff__eq,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( countable_countable(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
    <=> countable_countable(A,A3) ) ).

% countable_Diff_eq
tff(fact_7267_countable__image__eq__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image2(B,A,F2),S2))
    <=> ? [T10: set(B)] :
          ( countable_countable(B,T10)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T10),S2))
          & ( aa(set(B),set(A),image2(B,A,F2),S2) = aa(set(B),set(A),image2(B,A,F2),T10) )
          & inj_on(B,A,F2,T10) ) ) ).

% countable_image_eq_inj
tff(fact_7268_ex__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ? [T10: set(A)] :
          ( countable_countable(A,T10)
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T10),aa(set(B),set(A),image2(B,A,F2),S2)))
          & pp(aa(set(A),bool,P,T10)) )
    <=> ? [T10: set(B)] :
          ( countable_countable(B,T10)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T10),S2))
          & inj_on(B,A,F2,T10)
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F2),T10))) ) ) ).

% ex_countable_subset_image_inj
tff(fact_7269_all__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ! [T10: set(A)] :
          ( ( countable_countable(A,T10)
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T10),aa(set(B),set(A),image2(B,A,F2),S2))) )
         => pp(aa(set(A),bool,P,T10)) )
    <=> ! [T10: set(B)] :
          ( ( countable_countable(B,T10)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T10),S2))
            & inj_on(B,A,F2,T10) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F2),T10))) ) ) ).

% all_countable_subset_image_inj
tff(fact_7270_to__nat__on__surj,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( countable_countable(A,A3)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
       => ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
            & ( aa(A,nat,countable_to_nat_on(A,A3),X4) = N ) ) ) ) ).

% to_nat_on_surj
tff(fact_7271_uncountable__infinite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ countable_countable(A,A3)
     => ~ pp(aa(set(A),bool,finite_finite2(A),A3)) ) ).

% uncountable_infinite
tff(fact_7272_countable__finite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => countable_countable(A,S2) ) ).

% countable_finite
tff(fact_7273_countable__Collect__finite,axiom,
    ! [A: $tType] :
      ( countable(A)
     => countable_countable(set(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),finite_finite2(A))) ) ).

% countable_Collect_finite
tff(fact_7274_ccInf__less__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S2: set(A),A2: A] :
          ( countable_countable(A,S2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2))
          <=> ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),A2)) ) ) ) ) ).

% ccInf_less_iff
tff(fact_7275_less__ccSup__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S2: set(A),A2: A] :
          ( countable_countable(A,S2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2)))
          <=> ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X3)) ) ) ) ) ).

% less_ccSup_iff
tff(fact_7276_ccSup__subset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( countable_countable(A,B3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ).

% ccSup_subset_mono
tff(fact_7277_ccInf__superset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ) ).

% ccInf_superset_mono
tff(fact_7278_ccSup__upper2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),U: A,V2: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ) ).

% ccSup_upper2
tff(fact_7279_ccSup__le__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B2: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2))
          <=> ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2)) ) ) ) ) ).

% ccSup_le_iff
tff(fact_7280_ccSup__upper,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),X: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% ccSup_upper
tff(fact_7281_ccSup__least,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Z: A] :
          ( countable_countable(A,A3)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Z)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z)) ) ) ) ).

% ccSup_least
tff(fact_7282_ccSup__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( countable_countable(A,B3)
         => ( countable_countable(A,A3)
           => ( ! [A4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
                 => ? [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),X2)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ) ).

% ccSup_mono
tff(fact_7283_infinite__countable__subset_H,axiom,
    ! [A: $tType,X6: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),X6))
     => ? [C7: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C7),X6))
          & countable_countable(A,C7)
          & ~ pp(aa(set(A),bool,finite_finite2(A),C7)) ) ) ).

% infinite_countable_subset'
tff(fact_7284_countable__Collect__finite__subset,axiom,
    ! [A: $tType,T3: set(A)] :
      ( countable_countable(A,T3)
     => countable_countable(set(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_za(set(A),fun(set(A),bool),T3))) ) ).

% countable_Collect_finite_subset
tff(fact_7285_countable__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( countable_countable(A,B3)
       => countable_countable(A,A3) ) ) ).

% countable_subset
tff(fact_7286_ccInf__greatest,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Z: A] :
          ( countable_countable(A,A3)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X4)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A3))) ) ) ) ).

% ccInf_greatest
tff(fact_7287_le__ccInf__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B2: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A3)))
          <=> ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X3)) ) ) ) ) ).

% le_ccInf_iff
tff(fact_7288_ccInf__lower2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),U: A,V2: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),V2)) ) ) ) ) ).

% ccInf_lower2
tff(fact_7289_ccInf__lower,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),X: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),X)) ) ) ) ).

% ccInf_lower
tff(fact_7290_ccInf__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( countable_countable(A,B3)
         => ( countable_countable(A,A3)
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B3))
                 => ? [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),B4)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ) ) ).

% ccInf_mono
tff(fact_7291_all__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ! [T10: set(A)] :
          ( ( countable_countable(A,T10)
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T10),aa(set(B),set(A),image2(B,A,F2),S2))) )
         => pp(aa(set(A),bool,P,T10)) )
    <=> ! [T10: set(B)] :
          ( ( countable_countable(B,T10)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T10),S2)) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F2),T10))) ) ) ).

% all_countable_subset_image
tff(fact_7292_ex__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ? [T10: set(A)] :
          ( countable_countable(A,T10)
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T10),aa(set(B),set(A),image2(B,A,F2),S2)))
          & pp(aa(set(A),bool,P,T10)) )
    <=> ? [T10: set(B)] :
          ( countable_countable(B,T10)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T10),S2))
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image2(B,A,F2),T10))) ) ) ).

% ex_countable_subset_image
tff(fact_7293_countable__subset__image,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( ( countable_countable(A,B3)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3))) )
    <=> ? [A18: set(B)] :
          ( countable_countable(B,A18)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A18),A3))
          & ( B3 = aa(set(B),set(A),image2(B,A,F2),A18) ) ) ) ).

% countable_subset_image
tff(fact_7294_countable__image__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image2(B,A,F2),S2))
    <=> ? [T10: set(B)] :
          ( countable_countable(B,T10)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T10),S2))
          & ( aa(set(B),set(A),image2(B,A,F2),S2) = aa(set(B),set(A),image2(B,A,F2),T10) ) ) ) ).

% countable_image_eq
tff(fact_7295_uncountable__def,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ countable_countable(A,A3)
    <=> ( ( A3 != bot_bot(set(A)) )
        & ~ ? [F7: fun(nat,A)] : aa(set(nat),set(A),image2(nat,A,F7),top_top(set(nat))) = A3 ) ) ).

% uncountable_def
tff(fact_7296_ccSUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),B3: set(C),F2: fun(B,A),G: fun(C,A)] :
          ( countable_countable(B,A3)
         => ( countable_countable(C,B3)
           => ( ! [N2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N2),A3))
                 => ? [X2: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),B3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,N2)),aa(C,A,G,X2))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G),B3)))) ) ) ) ) ).

% ccSUP_mono
tff(fact_7297_ccSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),F2: fun(B,A),U: A] :
          ( countable_countable(B,A3)
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I2)),U)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),U)) ) ) ) ).

% ccSUP_least
tff(fact_7298_ccSUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I: B,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ) ).

% ccSUP_upper
tff(fact_7299_ccSUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),F2: fun(B,A),U: A] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),U))
          <=> ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X3)),U)) ) ) ) ) ).

% ccSUP_le_iff
tff(fact_7300_ccSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I: B,U: A,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,I)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ) ) ).

% ccSUP_upper2
tff(fact_7301_countable__infiniteE_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( countable_countable(A,A3)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
       => ~ ! [G3: fun(nat,A)] : ~ bij_betw(nat,A,G3,top_top(set(nat)),A3) ) ) ).

% countable_infiniteE'
tff(fact_7302_less__ccSUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [A3: set(B),A2: A,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))))
          <=> ? [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F2,X3))) ) ) ) ) ).

% less_ccSUP_iff
tff(fact_7303_ccINF__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),B3: set(C),F2: fun(B,A),G: fun(C,A)] :
          ( countable_countable(B,A3)
         => ( countable_countable(C,B3)
           => ( ! [M5: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),M5),B3))
                 => ? [X2: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X2)),aa(C,A,G,M5))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G),B3)))) ) ) ) ) ).

% ccINF_mono
tff(fact_7304_ccINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I: B,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(B,A,F2,I))) ) ) ) ).

% ccINF_lower
tff(fact_7305_ccINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I: B,F2: fun(B,A),U: A] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,I)),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),U)) ) ) ) ) ).

% ccINF_lower2
tff(fact_7306_le__ccINF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),U: A,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))))
          <=> ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,X3))) ) ) ) ) ).

% le_ccINF_iff
tff(fact_7307_ccINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),U: A,F2: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F2,I2))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)))) ) ) ) ).

% ccINF_greatest
tff(fact_7308_ccINF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [A3: set(B),F2: fun(B,A),A2: A] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),A2))
          <=> ? [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,X3)),A2)) ) ) ) ) ).

% ccINF_less_iff
tff(fact_7309_countableE__infinite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
       => ~ ! [E2: fun(A,nat)] : ~ bij_betw(A,nat,E2,S2,top_top(set(nat))) ) ) ).

% countableE_infinite
tff(fact_7310_ccSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( countable_countable(A,A3)
         => ( countable_countable(A,B3)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)))) ) ) ) ).

% ccSup_inter_less_eq
tff(fact_7311_less__eq__ccInf__inter,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( countable_countable(A,A3)
         => ( countable_countable(A,B3)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))) ) ) ) ).

% less_eq_ccInf_inter
tff(fact_7312_countable__vimage,axiom,
    ! [B: $tType,A: $tType,B3: set(A),F2: fun(B,A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))))
     => ( countable_countable(B,vimage(B,A,F2,B3))
       => countable_countable(A,B3) ) ) ).

% countable_vimage
tff(fact_7313_ccSUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(B),A3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( countable_countable(B,B3)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,G,X4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),B3)))) ) ) ) ) ).

% ccSUP_subset_mono
tff(fact_7314_ccINF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),B3: set(B),F2: fun(B,A),G: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),B3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F2,X4)),aa(B,A,G,X4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),B3)))) ) ) ) ) ).

% ccINF_superset_mono
tff(fact_7315_mono__ccSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),I6: set(C),A3: fun(C,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( countable_countable(C,I6)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aii(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I6))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A3),I6))))) ) ) ) ).

% mono_ccSUP
tff(fact_7316_mono__ccSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( countable_countable(A,A3)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3)))) ) ) ) ).

% mono_ccSup
tff(fact_7317_mono__ccINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta4013691401010221786attice(A) )
     => ! [F2: fun(A,B),I6: set(C),A3: fun(C,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( countable_countable(C,I6)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A3),I6)))),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_aii(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I6)))) ) ) ) ).

% mono_ccINF
tff(fact_7318_mono__ccInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( countable_countable(A,A3)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3)))) ) ) ) ).

% mono_ccInf
tff(fact_7319_countable__as__injective__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( countable_countable(A,A3)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
       => ~ ! [F3: fun(nat,A)] :
              ( ( A3 = aa(set(nat),set(A),image2(nat,A,F3),top_top(set(nat))) )
             => ~ inj_on(nat,A,F3,top_top(set(nat))) ) ) ) ).

% countable_as_injective_image
tff(fact_7320_image__to__nat__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( countable_countable(A,A3)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
       => ( aa(set(A),set(nat),image2(A,nat,countable_to_nat_on(A,A3)),A3) = top_top(set(nat)) ) ) ) ).

% image_to_nat_on
tff(fact_7321_to__nat__on__infinite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
       => bij_betw(A,nat,countable_to_nat_on(A,S2),S2,top_top(set(nat))) ) ) ).

% to_nat_on_infinite
tff(fact_7322_countable__enum__cases,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ! [F3: fun(A,nat)] : ~ bij_betw(A,nat,F3,S2,set_ord_lessThan(nat,aa(set(A),nat,finite_card(A),S2))) )
       => ~ ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
           => ! [F3: fun(A,nat)] : ~ bij_betw(A,nat,F3,S2,top_top(set(nat))) ) ) ) ).

% countable_enum_cases
tff(fact_7323_range__from__nat__into,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( countable_countable(A,A3)
       => ( aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3)),top_top(set(nat))) = A3 ) ) ) ).

% range_from_nat_into
tff(fact_7324_vanishes__mult__bounded,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( ? [A10: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A10))
          & ! [N2: nat] : pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N2))),A10)) )
     => ( vanishes(Y5)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aij(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X6),Y5)) ) ) ).

% vanishes_mult_bounded
tff(fact_7325_from__nat__into__inject,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( countable_countable(A,A3)
       => ( ( B3 != bot_bot(set(A)) )
         => ( countable_countable(A,B3)
           => ( ( aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3) = aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),B3) )
            <=> ( A3 = B3 ) ) ) ) ) ) ).

% from_nat_into_inject
tff(fact_7326_from__nat__into__inj__infinite,axiom,
    ! [A: $tType,A3: set(A),M: nat,N: nat] :
      ( countable_countable(A,A3)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
       => ( ( aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3),M) = aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3),N) )
        <=> ( M = N ) ) ) ) ).

% from_nat_into_inj_infinite
tff(fact_7327_to__nat__on__from__nat__into__infinite,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( countable_countable(A,A3)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
       => ( aa(A,nat,countable_to_nat_on(A,A3),aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3),N)) = N ) ) ) ).

% to_nat_on_from_nat_into_infinite
tff(fact_7328_from__nat__into,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( ( A3 != bot_bot(set(A)) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3),N)),A3)) ) ).

% from_nat_into
tff(fact_7329_inj__on__from__nat__into,axiom,
    ! [A: $tType] : inj_on(set(A),fun(nat,A),counta4804993851260445106t_into(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_aik(set(A),bool))) ).

% inj_on_from_nat_into
tff(fact_7330_vanishes__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( vanishes(X6)
    <=> ! [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
         => ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N3))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N3))),R5)) ) ) ) ).

% vanishes_def
tff(fact_7331_vanishesI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3))
         => ? [K8: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K8),N2))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N2))),R3)) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_7332_vanishesD,axiom,
    ! [X6: fun(nat,rat),R2: rat] :
      ( vanishes(X6)
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R2))
       => ? [K2: nat] :
          ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N5))
           => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N5))),R2)) ) ) ) ).

% vanishesD
tff(fact_7333_range__from__nat__into__subset,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3)),top_top(set(nat)))),A3)) ) ).

% range_from_nat_into_subset
tff(fact_7334_subset__range__from__nat__into,axiom,
    ! [A: $tType,A3: set(A)] :
      ( countable_countable(A,A3)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3)),top_top(set(nat))))) ) ).

% subset_range_from_nat_into
tff(fact_7335_bij__betw__from__nat__into__finite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => bij_betw(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),S2),set_ord_lessThan(nat,aa(set(A),nat,finite_card(A),S2)),S2) ) ).

% bij_betw_from_nat_into_finite
tff(fact_7336_bij__betw__from__nat__into,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
       => bij_betw(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),S2),top_top(set(nat)),S2) ) ) ).

% bij_betw_from_nat_into
tff(fact_7337_butlast__power,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(nat,fun(fun(list(A),list(A)),fun(list(A),list(A))),compow(fun(list(A),list(A))),N),butlast(A)),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)),Xs) ).

% butlast_power
tff(fact_7338_comp__fun__idem__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
    <=> ( finite4664212375090638736ute_on(A,B,S2,F2)
        & finite4980608107308702382axioms(A,B,S2,F2) ) ) ).

% comp_fun_idem_on_def
tff(fact_7339_butlast__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),Xs)) ).

% butlast_rev
tff(fact_7340_butlast__snoc,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = Xs ).

% butlast_snoc
tff(fact_7341_length__butlast,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_butlast
tff(fact_7342_comp__fun__idem__on__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X4)),aa(A,fun(B,B),F2,X4)) = aa(A,fun(B,B),F2,X4) ) )
     => finite4980608107308702382axioms(A,B,S2,F2) ) ).

% comp_fun_idem_on_axioms.intro
tff(fact_7343_comp__fun__idem__on__axioms__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite4980608107308702382axioms(A,B,S2,F2)
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
         => ( 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) ) ) ) ).

% comp_fun_idem_on_axioms_def
tff(fact_7344_in__set__butlast__appendI,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs))))
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Ys)))) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))))) ) ).

% in_set_butlast_appendI
tff(fact_7345_in__set__butlastD,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_butlastD
tff(fact_7346_distinct__butlast,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),butlast(A),Xs)) ) ).

% distinct_butlast
tff(fact_7347_drop__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),aa(list(A),list(A),butlast(A),Xs)) = aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) ).

% drop_butlast
tff(fact_7348_butlast_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),butlast(A),Xs)) ) ) ) ).

% butlast.simps(2)
tff(fact_7349_butlast_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),butlast(A),nil(A)) = nil(A) ).

% butlast.simps(1)
tff(fact_7350_butlast__append,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( ( ( Ys = nil(A) )
       => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),butlast(A),Xs) ) )
      & ( ( Ys != nil(A) )
       => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),butlast(A),Ys)) ) ) ) ).

% butlast_append
tff(fact_7351_butlast__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),aa(list(A),list(A),tl(A),Xs)) = aa(list(A),list(A),tl(A),aa(list(A),list(A),butlast(A),Xs)) ).

% butlast_tl
tff(fact_7352_map__butlast,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),aa(list(B),list(B),butlast(B),Xs)) = aa(list(A),list(A),butlast(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs)) ).

% map_butlast
tff(fact_7353_comp__fun__idem__on_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => finite4980608107308702382axioms(A,B,S2,F2) ) ).

% comp_fun_idem_on.axioms(2)
tff(fact_7354_nth__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xs))))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),butlast(A),Xs)),N) = aa(nat,A,nth(A,Xs),N) ) ) ).

% nth_butlast
tff(fact_7355_sorted__butlast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),butlast(A),Xs)) ) ) ) ).

% sorted_butlast
tff(fact_7356_take__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),aa(list(A),list(A),butlast(A),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs) ) ) ).

% take_butlast
tff(fact_7357_butlast__conv__take,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),Xs) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))),Xs) ).

% butlast_conv_take
tff(fact_7358_butlast__list__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),X: A] :
      ( ( ( K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
       => ( aa(list(A),list(A),butlast(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X)) = aa(list(A),list(A),butlast(A),Xs) ) )
      & ( ( K != aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
       => ( aa(list(A),list(A),butlast(A),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),aa(list(A),list(A),butlast(A),Xs)),K),X) ) ) ) ).

% butlast_list_update
tff(fact_7359_butlast__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),Xs) ) ) ).

% butlast_take
tff(fact_7360_comp__fun__idem__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( finite4980608107308702382axioms(A,B,S2,F2)
       => finite673082921795544331dem_on(A,B,S2,F2) ) ) ).

% comp_fun_idem_on.intro
tff(fact_7361_and__not__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa2))
       => ( ( ( X = one2 )
           => ( ( Xa2 = one2 )
             => ( ( Y = none(num) )
               => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa2 = aa(num,num,bit0,N2) )
                 => ( ( Y = aa(num,option(num),some(num),one2) )
                   => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N2))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa2 = aa(num,num,bit1,N2) )
                   => ( ( Y = none(num) )
                     => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N2))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa2 = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                       => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit0,N2) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) )
                           => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N2))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N2: num] :
                            ( ( Xa2 = aa(num,num,bit1,N2) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) )
                             => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N2))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa2 = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                             => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one2)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit0,N2) )
                               => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_abu(num,option(num)),bit_and_not_num(M5,N2)) )
                                 => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N2))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N2: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N2) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) )
                                   => ~ accp(product_prod(num,num),bit_and_not_num_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_7362_shuffles_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa2) = Y )
     => ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa2))
       => ( ( ( X = nil(A) )
           => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xa2),bot_bot(set(list(A)))) )
             => ~ accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa2)) ) )
         => ( ( ( Xa2 = nil(A) )
             => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),X),bot_bot(set(list(A)))) )
               => ~ accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),nil(A))) ) )
           => ~ ! [X4: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
                 => ! [Y2: A,Ys3: list(A)] :
                      ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
                     => ( ( Y = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2),Ys3))) )
                       => ~ accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_7363_shuffles_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1))
     => ( ! [Ys3: list(A)] :
            ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys3)) )
       => ( ! [Xs2: list(A)] :
              ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A)))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),nil(A))) )
         => ( ! [X4: A,Xs2: list(A),Y2: A,Ys3: list(A)] :
                ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)))
               => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)))
                 => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Ys3))
                   => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3))) ) ) )
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ) ).

% shuffles.pinduct
tff(fact_7364_shuffles_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys))
     => ( shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Ys),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(1)
tff(fact_7365_shuffles_Opsimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)))
     => ( shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xs),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(2)
tff(fact_7366_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A)] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))
     => ( shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys))) ) ) ).

% shuffles.psimps(3)
tff(fact_7367_and__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa2))
       => ( ( ( X = one2 )
           => ( ( Xa2 = one2 )
             => ( ( Y = aa(num,option(num),some(num),one2) )
               => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa2 = aa(num,num,bit0,N2) )
                 => ( ( Y = none(num) )
                   => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N2))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa2 = aa(num,num,bit1,N2) )
                   => ( ( Y = aa(num,option(num),some(num),one2) )
                     => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N2))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa2 = one2 )
                     => ( ( Y = none(num) )
                       => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit0,N2) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N2)) )
                           => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N2))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N2: num] :
                            ( ( Xa2 = aa(num,num,bit1,N2) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N2)) )
                             => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N2))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa2 = one2 )
                           => ( ( Y = aa(num,option(num),some(num),one2) )
                             => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one2)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit0,N2) )
                               => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N2)) )
                                 => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N2))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N2: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N2) )
                                 => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_abu(num,option(num)),bit_un7362597486090784418nd_num(M5,N2)) )
                                   => ~ accp(product_prod(num,num),bit_un4731106466462545111um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_7368_xor__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un2480387367778600638or_num(X,Xa2) = Y )
     => ( accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa2))
       => ( ( ( X = one2 )
           => ( ( Xa2 = one2 )
             => ( ( Y = none(num) )
               => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa2 = aa(num,num,bit0,N2) )
                 => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,N2)) )
                   => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N2))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa2 = aa(num,num,bit1,N2) )
                   => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,N2)) )
                     => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N2))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa2 = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,M5)) )
                       => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa2 = aa(num,num,bit0,N2) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M5,N2)) )
                           => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N2))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N2: num] :
                            ( ( Xa2 = aa(num,num,bit1,N2) )
                           => ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M5,N2))) )
                             => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N2))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa2 = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                             => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one2)) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa2 = aa(num,num,bit0,N2) )
                               => ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_un2480387367778600638or_num(M5,N2))) )
                                 => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N2))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N2: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N2) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un2480387367778600638or_num(M5,N2)) )
                                   => ~ accp(product_prod(num,num),bit_un2901131394128224187um_rel,aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_7369_splice_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
      ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1))
     => ( ! [Ys3: list(A)] :
            ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys3)) )
       => ( ! [X4: A,Xs2: list(A),Ys3: list(A)] :
              ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Ys3))
             => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Ys3),Xs2))
               => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Ys3)) ) )
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ).

% splice.pinduct
tff(fact_7370_Field__insert,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R2)) ).

% Field_insert
tff(fact_7371_Field__empty,axiom,
    ! [A: $tType] : field2(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(A)) ).

% Field_empty
tff(fact_7372_mono__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),field2(A,R2)),field2(A,S))) ) ).

% mono_Field
tff(fact_7373_Order__Relation_OunderS__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R2,A2)),field2(A,R2))) ).

% Order_Relation.underS_Field
tff(fact_7374_underS__empty,axiom,
    ! [A: $tType,A2: A,R2: set(product_prod(A,A))] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
     => ( order_underS(A,R2,A2) = bot_bot(set(A)) ) ) ).

% underS_empty
tff(fact_7375_underS__Field2,axiom,
    ! [A: $tType,A2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),order_underS(A,R2,A2)),field2(A,R2))) ) ).

% underS_Field2
tff(fact_7376_finite__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => pp(aa(set(A),bool,finite_finite2(A),field2(A,R2))) ) ).

% finite_Field
tff(fact_7377_FieldI2,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J),field2(A,R))) ) ).

% FieldI2
tff(fact_7378_FieldI1,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),field2(A,R))) ) ).

% FieldI1
tff(fact_7379_underS__Field3,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] :
      ( ( field2(A,R2) != bot_bot(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),order_underS(A,R2,A2)),field2(A,R2))) ) ).

% underS_Field3
tff(fact_7380_Field__Restr__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(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,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3))))),A3)) ).

% Field_Restr_subset
tff(fact_7381_Field__natLeq__on,axiom,
    ! [N: nat] : field2(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ahh(nat,fun(nat,fun(nat,bool)),N)))) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),N)) ).

% Field_natLeq_on
tff(fact_7382_underS__incl__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R2,A2)),order_underS(A,R2,B2)))
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) ) ) ) ) ).

% underS_incl_iff
tff(fact_7383_Refl__Field__Restr2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] :
      ( refl_on(A,field2(A,R2),R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),field2(A,R2)))
       => ( field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),A3)))) = A3 ) ) ) ).

% Refl_Field_Restr2
tff(fact_7384_Under__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] : order_Under(A,R2,A3) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ail(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R2),A3)) ).

% Under_def
tff(fact_7385_UnderS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] : order_UnderS(A,R2,A3) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aim(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R2),A3)) ).

% UnderS_def
tff(fact_7386_Above__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] : order_Above(A,R2,A3) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ain(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R2),A3)) ).

% Above_def
tff(fact_7387_finite__Linear__order__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,P: fun(A,bool)] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),field2(A,R2)))
       => ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),field2(A,R2)))
               => ( ! [Y3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),order_aboveS(A,R2,X4)))
                     => pp(aa(A,bool,P,Y3)) )
                 => pp(aa(A,bool,P,X4)) ) )
           => pp(aa(A,bool,P,X)) ) ) ) ) ).

% finite_Linear_order_induct
tff(fact_7388_aboveS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : order_aboveS(A,R2,A2) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aio(set(product_prod(A,A)),fun(A,fun(A,bool)),R2),A2)) ).

% aboveS_def
tff(fact_7389_splice_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),X),Xa2) = Y )
     => ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Xa2))
       => ( ( ( X = nil(A) )
           => ( ( Y = Xa2 )
             => ~ accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa2)) ) )
         => ~ ! [X4: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
               => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xa2),Xs2)) )
                 => ~ accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),Xa2)) ) ) ) ) ) ).

% splice.pelims
tff(fact_7390_cofinal__def,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( bNF_Ca7293521722713021262ofinal(A,A3,R2)
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),field2(A,R2)))
         => ? [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A3))
              & ( X3 != Xa3 )
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),R2)) ) ) ) ).

% cofinal_def
tff(fact_7391_splice__Nil2,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xs),nil(A)) = Xs ).

% splice_Nil2
tff(fact_7392_split__Nil__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xs),Ys) = nil(A) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% split_Nil_iff
tff(fact_7393_splice__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xs),Ys)),shuffles(A,Xs,Ys))) ).

% splice_in_shuffles
tff(fact_7394_length__splice,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_splice
tff(fact_7395_splice__replicate,axiom,
    ! [A: $tType,M: nat,X: A,N: nat] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),M),X)),aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),X) ).

% splice_replicate
tff(fact_7396_splice_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),nil(A)),Ys) = Ys ).

% splice.simps(1)
tff(fact_7397_splice_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Ys),Xs)) ).

% splice.simps(2)
tff(fact_7398_splice_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),X),Xa2) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != Xa2 ) )
       => ~ ! [X4: A,Xs2: list(A)] :
              ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
             => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Xa2),Xs2)) ) ) ) ) ).

% splice.elims
tff(fact_7399_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
      ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),Ys),Xs)) ) ) ).

% splice.psimps(2)
tff(fact_7400_splice_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),splice(A),nil(A)),Ys) = Ys ) ) ).

% splice.psimps(1)
tff(fact_7401_Refl__under__underS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] :
      ( refl_on(A,field2(A,R2),R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( order_under(A,R2,A2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),order_underS(A,R2,A2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ) ) ).

% Refl_under_underS
tff(fact_7402_Total__subset__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R2),R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),id(A)))
       => ( ( R2 = bot_bot(set(product_prod(A,A))) )
          | ? [A4: 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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_7403_IdI,axiom,
    ! [A: $tType,A2: A] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2)),id(A))) ).

% IdI
tff(fact_7404_pair__in__Id__conv,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),id(A)))
    <=> ( A2 = B2 ) ) ).

% pair_in_Id_conv
tff(fact_7405_rtrancl__empty,axiom,
    ! [A: $tType] : transitive_rtrancl(A,bot_bot(set(product_prod(A,A)))) = id(A) ).

% rtrancl_empty
tff(fact_7406_under__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_under(A,R2,A2)),field2(A,R2))) ).

% under_Field
tff(fact_7407_underS__subset__under,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R2,A2)),order_under(A,R2,A2))) ).

% underS_subset_under
tff(fact_7408_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) = id(A) ).

% relpow.simps(1)
tff(fact_7409_IdE,axiom,
    ! [A: $tType,P2: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P2),id(A)))
     => ~ ! [X4: A] : P2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).

% IdE
tff(fact_7410_IdD,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),id(A)))
     => ( A2 = B2 ) ) ).

% IdD
tff(fact_7411_under__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : order_under(A,R2,A2) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aip(set(product_prod(A,A)),fun(A,fun(A,bool)),R2),A2)) ).

% under_def
tff(fact_7412_Id__def,axiom,
    ! [A: $tType] : id(A) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aiq(product_prod(A,A),bool)) ).

% Id_def
tff(fact_7413_Linear__order__in__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id(A)))) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_7414_reflcl__set__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X2: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),aTP_Lamp_wl(set(product_prod(A,A)),fun(A,fun(A,bool)),R2)),fequal(A)),X2),Xa))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),id(A)))) ) ).

% reflcl_set_eq
tff(fact_7415_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)),bool),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),bool)),fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),product_case_prod(product_prod(A,A),product_prod(A,A),bool),aa(fun(A,fun(A,fun(product_prod(A,A),bool))),fun(product_prod(A,A),fun(product_prod(A,A),bool)),product_case_prod(A,A,fun(product_prod(A,A),bool)),aTP_Lamp_ais(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),R2)))) ).

% bsqr_def
tff(fact_7416_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B,Y: B,A2: A] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),Y))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [Y8: B] :
                ( ( Y = aa(B,B,aa(A,fun(B,B),F2,A2),Y8) )
                & pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))),Y8)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_7417_finite__imp__fold__graph,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ? [X_1: B] : pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),X_1)) ) ).

% finite_imp_fold_graph
tff(fact_7418_fold__graph__closed__lemma,axiom,
    ! [A: $tType,B: $tType,G: fun(A,fun(B,B)),Z: B,A3: set(A),X: B,B3: set(B),F2: fun(A,fun(B,B))] :
      ( pp(aa(B,bool,finite_fold_graph(A,B,G,Z,A3),X))
     => ( ! [A4: A,B4: B] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),B3))
             => ( aa(B,B,aa(A,fun(B,B),F2,A4),B4) = aa(B,B,aa(A,fun(B,B),G,A4),B4) ) ) )
       => ( ! [A4: A,B4: B] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),B3))
               => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(B,B,aa(A,fun(B,B),G,A4),B4)),B3)) ) )
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Z),B3))
           => ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),X))
              & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),B3)) ) ) ) ) ) ).

% fold_graph_closed_lemma
tff(fact_7419_fold__graph__closed__eq,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),F2: fun(A,fun(B,B)),G: fun(A,fun(B,B)),Z: B] :
      ( ! [A4: A,B4: B] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),B3))
           => ( aa(B,B,aa(A,fun(B,B),F2,A4),B4) = aa(B,B,aa(A,fun(B,B),G,A4),B4) ) ) )
     => ( ! [A4: A,B4: B] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A4),A3))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),B3))
             => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(B,B,aa(A,fun(B,B),G,A4),B4)),B3)) ) )
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Z),B3))
         => ( finite_fold_graph(A,B,F2,Z,A3) = finite_fold_graph(A,B,G,Z,A3) ) ) ) ) ).

% fold_graph_closed_eq
tff(fact_7420_empty__fold__graphE,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B,X: B] :
      ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,bot_bot(set(A))),X))
     => ( X = Z ) ) ).

% empty_fold_graphE
tff(fact_7421_fold__graph_OemptyI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B] : pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,bot_bot(set(A))),Z)) ).

% fold_graph.emptyI
tff(fact_7422_comp__fun__commute__on_Ofold__graph__finite,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Z: B,A3: set(A),Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),Y))
       => pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ).

% comp_fun_commute_on.fold_graph_finite
tff(fact_7423_fold__graph_OinsertI,axiom,
    ! [A: $tType,B: $tType,X: A,A3: set(A),F2: fun(A,fun(B,B)),Z: B,Y: B] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),Y))
       => pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),aa(B,B,aa(A,fun(B,B),F2,X),Y))) ) ) ).

% fold_graph.insertI
tff(fact_7424_fold__graph_Ocases,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B,A1: set(A),A22: B] :
      ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A1),A22))
     => ( ( ( A1 = bot_bot(set(A)) )
         => ( A22 != Z ) )
       => ~ ! [X4: A,A5: set(A)] :
              ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),A5) )
             => ! [Y2: B] :
                  ( ( A22 = aa(B,B,aa(A,fun(B,B),F2,X4),Y2) )
                 => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A5))
                   => ~ pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A5),Y2)) ) ) ) ) ) ).

% fold_graph.cases
tff(fact_7425_fold__graph_Osimps,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B,A1: set(A),A22: B] :
      ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A1),A22))
    <=> ( ( ( A1 = bot_bot(set(A)) )
          & ( A22 = Z ) )
        | ? [X3: A,A8: set(A),Y4: B] :
            ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),A8) )
            & ( A22 = aa(B,B,aa(A,fun(B,B),F2,X3),Y4) )
            & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A8))
            & pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A8),Y4)) ) ) ) ).

% fold_graph.simps
tff(fact_7426_comp__fun__commute__on_Ofold__graph__determ,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B,X: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),X))
         => ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),Y))
           => ( Y = X ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_determ
tff(fact_7427_fold__graph__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A3: set(A),F2: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G,A3)
     => ( finite_fold_graph(B,C,F2,Z,aa(set(A),set(B),image2(A,B,G),A3)) = finite_fold_graph(A,C,aa(fun(A,B),fun(A,fun(C,C)),comp(B,fun(C,C),A,F2),G),Z,A3) ) ) ).

% fold_graph_image
tff(fact_7428_comp__fun__commute__on_Ofold__graph__insertE,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),X: A,A3: set(A),Z: B,V2: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),S2))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A3)),V2))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ~ ! [Y2: B] :
                  ( ( V2 = aa(B,B,aa(A,fun(B,B),F2,X),Y2) )
                 => ~ pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),Y2)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE
tff(fact_7429_comp__fun__commute__on_Ofold__equality,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),Y))
         => ( finite_fold(A,B,F2,Z,A3) = Y ) ) ) ) ).

% comp_fun_commute_on.fold_equality
tff(fact_7430_Finite__Set_Ofold__def,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),A3))
       => ( finite_fold(A,B,F2,Z,A3) = the(B,finite_fold_graph(A,B,F2,Z,A3)) ) )
      & ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
       => ( finite_fold(A,B,F2,Z,A3) = Z ) ) ) ).

% Finite_Set.fold_def
tff(fact_7431_comp__fun__commute__on_Ofold__graph__fold,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => pp(aa(B,bool,finite_fold_graph(A,B,F2,Z,A3),finite_fold(A,B,F2,Z,A3))) ) ) ) ).

% comp_fun_commute_on.fold_graph_fold
tff(fact_7432_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id(A)))
      <=> ! [A8: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),field2(A,R2)))
           => ( ( A8 != bot_bot(set(A)) )
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A8))
                  & ! [Xa3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A8))
                     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),R2)) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_7433_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A2: list(A),B2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),A2),B2))
    <=> ? [Z2: list(product_prod(A,B))] :
          ( pp(aa(set(list(product_prod(A,B))),bool,aa(list(product_prod(A,B)),fun(set(list(product_prod(A,B))),bool),member(list(product_prod(A,B))),Z2),aa(fun(list(product_prod(A,B)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ait(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R))))
          & ( aa(list(product_prod(A,B)),list(A),aa(fun(product_prod(A,B),A),fun(list(product_prod(A,B)),list(A)),map(product_prod(A,B),A),product_fst(A,B)),Z2) = A2 )
          & ( aa(list(product_prod(A,B)),list(B),aa(fun(product_prod(A,B),B),fun(list(product_prod(A,B)),list(B)),map(product_prod(A,B),B),product_snd(A,B)),Z2) = B2 ) ) ) ).

% list.in_rel
tff(fact_7434_list__all2__Nil2,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Xs: list(A)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),nil(B)))
    <=> ( Xs = nil(A) ) ) ).

% list_all2_Nil2
tff(fact_7435_list__all2__Nil,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),nil(A)),Ys))
    <=> ( Ys = nil(B) ) ) ).

% list_all2_Nil
tff(fact_7436_list__all2__rev,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),rev(A),Xs)),aa(list(B),list(B),rev(B),Ys)))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys)) ) ).

% list_all2_rev
tff(fact_7437_wf__empty,axiom,
    ! [A: $tType] : wf(A,bot_bot(set(product_prod(A,A)))) ).

% wf_empty
tff(fact_7438_wf__listrel1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(list(A),listrel1(A,R2))
    <=> wf(A,R2) ) ).

% wf_listrel1_iff
tff(fact_7439_wf__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lex(A,R2)) ) ).

% wf_lex
tff(fact_7440_wf__lenlex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lenlex(A,R2)) ) ).

% wf_lenlex
tff(fact_7441_wf__insert,axiom,
    ! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
      ( wf(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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2))
    <=> ( wf(A,R2)
        & ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2))) ) ) ).

% wf_insert
tff(fact_7442_reduction__pairI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( wf(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S2)),R))
       => fun_reduction_pair(A,aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),S2)) ) ) ).

% reduction_pairI
tff(fact_7443_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P: fun(B,bool),K: B,M: fun(B,A)] :
      ( wf(A,R2)
     => ( ! [X4: A,Y2: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y2)),transitive_trancl(A,R2)))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X4)),transitive_rtrancl(A,R2))) )
       => ( pp(aa(B,bool,P,K))
         => ? [X4: B] :
              ( pp(aa(B,bool,P,X4))
              & ! [Y3: B] :
                  ( pp(aa(B,bool,P,Y3))
                 => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M,X4)),aa(B,A,M,Y3))),transitive_rtrancl(A,R2))) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_7444_list_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R),Ra2))
     => pp(aa(fun(list(A),fun(list(B),bool)),bool,aa(fun(list(A),fun(list(B),bool)),fun(fun(list(A),fun(list(B),bool)),bool),ord_less_eq(fun(list(A),fun(list(B),bool))),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),Ra2))) ) ).

% list.rel_mono
tff(fact_7445_wf__bounded__measure,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,nat),F2: fun(A,nat)] :
      ( ! [A4: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4)),R2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Ub,B4)),aa(A,nat,Ub,A4)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,B4)),aa(A,nat,Ub,A4)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,A4)),aa(A,nat,F2,B4))) ) )
     => wf(A,R2) ) ).

% wf_bounded_measure
tff(fact_7446_wf,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),ord_less(A)))) ) ).

% wf
tff(fact_7447_wf__if__measure,axiom,
    ! [A: $tType,P: fun(A,bool),F2: fun(A,nat),G: fun(A,A)] :
      ( ! [X4: A] :
          ( pp(aa(A,bool,P,X4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,aa(A,A,G,X4))),aa(A,nat,F2,X4))) )
     => wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_aiu(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),P),G)))) ) ).

% wf_if_measure
tff(fact_7448_wf__less,axiom,
    wf(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less(nat)))) ).

% wf_less
tff(fact_7449_list_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: list(A),G: fun(B,C),Y: list(B)] :
      ( pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),Sa),X),aa(list(B),list(C),aa(fun(B,C),fun(list(B),list(C)),map(B,C),G),Y)))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_aiv(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y)) ) ).

% list.rel_map(2)
tff(fact_7450_list_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X: list(A),Y: list(B)] :
      ( pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),aa(fun(C,fun(B,bool)),fun(list(C),fun(list(B),bool)),list_all2(C,B),Sb),aa(list(A),list(C),aa(fun(A,C),fun(list(A),list(C)),map(A,C),I),X)),Y))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_aiw(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X),Y)) ) ).

% list.rel_map(1)
tff(fact_7451_list__all2__map1,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),F2: fun(C,A),As: list(C),Bs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(C),list(A),aa(fun(C,A),fun(list(C),list(A)),map(C,A),F2),As)),Bs))
    <=> pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),aa(fun(C,fun(B,bool)),fun(list(C),fun(list(B),bool)),list_all2(C,B),aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_aix(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),P),F2)),As),Bs)) ) ).

% list_all2_map1
tff(fact_7452_list__all2__map2,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,bool)),As: list(A),F2: fun(C,B),Bs: list(C)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),As),aa(list(C),list(B),aa(fun(C,B),fun(list(C),list(B)),map(C,B),F2),Bs)))
    <=> pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_aiy(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),P),F2)),As),Bs)) ) ).

% list_all2_map2
tff(fact_7453_list__all2__appendI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),A2: list(A),B2: list(B),C2: list(A),D2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),A2),B2))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),C2),D2))
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),C2)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),B2),D2))) ) ) ).

% list_all2_appendI
tff(fact_7454_list_Orel__inject_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X21: A,X222: list(A),Y21: B,Y222: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y222)))
    <=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,X21),Y21))
        & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X222),Y222)) ) ) ).

% list.rel_inject(2)
tff(fact_7455_list_Orel__intros_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X21: A,Y21: B,X222: list(A),Y222: list(B)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R,X21),Y21))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X222),Y222))
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y222))) ) ) ).

% list.rel_intros(2)
tff(fact_7456_list__all2__Cons,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Xs: list(A),Y: B,Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)))
    <=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
        & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys)) ) ) ).

% list_all2_Cons
tff(fact_7457_list__all2__Cons1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys))
    <=> ? [Z2: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z2),Zs3) )
          & pp(aa(B,bool,aa(A,fun(B,bool),P,X),Z2))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Zs3)) ) ) ).

% list_all2_Cons1
tff(fact_7458_list__all2__Cons2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Y: B,Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)))
    <=> ? [Z2: A,Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs3) )
          & pp(aa(B,bool,aa(A,fun(B,bool),P,Z2),Y))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Zs3),Ys)) ) ) ).

% list_all2_Cons2
tff(fact_7459_wf__no__loop,axiom,
    ! [B: $tType,R: set(product_prod(B,B))] :
      ( ( relcomp(B,B,B,R,R) = bot_bot(set(product_prod(B,B))) )
     => wf(B,R) ) ).

% wf_no_loop
tff(fact_7460_list_Orel__eq,axiom,
    ! [A: $tType] : aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),fequal(A)) = fequal(list(A)) ).

% list.rel_eq
tff(fact_7461_list_Orel__refl,axiom,
    ! [B: $tType,Ra2: fun(B,fun(B,bool)),X: list(B)] :
      ( ! [X4: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra2,X4),X4))
     => pp(aa(list(B),bool,aa(list(B),fun(list(B),bool),aa(fun(B,fun(B,bool)),fun(list(B),fun(list(B),bool)),list_all2(B,B),Ra2),X),X)) ) ).

% list.rel_refl
tff(fact_7462_list__all2__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),fequal(A)),Xs),Ys)) ) ).

% list_all2_eq
tff(fact_7463_list__all2__mono,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),Q: fun(A,fun(B,bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
     => ( ! [Xs2: A,Ys3: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),P,Xs2),Ys3))
           => pp(aa(B,bool,aa(A,fun(B,bool),Q,Xs2),Ys3)) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),Q),Xs),Ys)) ) ) ).

% list_all2_mono
tff(fact_7464_list__all2__refl,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( ! [X4: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,X4),X4))
     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),P),Xs),Xs)) ) ).

% list_all2_refl
tff(fact_7465_list__all2__trans,axiom,
    ! [B: $tType,A: $tType,C: $tType,P1: fun(A,fun(B,bool)),P22: fun(B,fun(C,bool)),P32: fun(A,fun(C,bool)),As: list(A),Bs: list(B),Cs: list(C)] :
      ( ! [A4: A,B4: B,C4: C] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),P1,A4),B4))
         => ( pp(aa(C,bool,aa(B,fun(C,bool),P22,B4),C4))
           => pp(aa(C,bool,aa(A,fun(C,bool),P32,A4),C4)) ) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P1),As),Bs))
       => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(fun(B,fun(C,bool)),fun(list(B),fun(list(C),bool)),list_all2(B,C),P22),Bs),Cs))
         => pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),P32),As),Cs)) ) ) ) ).

% list_all2_trans
tff(fact_7466_list__all2__antisym,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Q: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Y2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),Q,Y2),X4))
           => ( X4 = Y2 ) ) )
     => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),P),Xs),Ys))
       => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),Q),Ys),Xs))
         => ( Xs = Ys ) ) ) ) ).

% list_all2_antisym
tff(fact_7467_list__all2__update__cong,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),X: A,Y: B,I: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),I),X)),aa(B,list(B),aa(nat,fun(B,list(B)),aa(list(B),fun(nat,fun(B,list(B))),list_update(B),Ys),I),Y))) ) ) ).

% list_all2_update_cong
tff(fact_7468_wf__lexn,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] :
      ( wf(A,R2)
     => wf(list(A),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N)) ) ).

% wf_lexn
tff(fact_7469_list_Octr__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),nil(A)),nil(B))) ).

% list.ctr_transfer(1)
tff(fact_7470_list__all2__induct,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),R: fun(list(A),fun(list(B),bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,nil(A)),nil(B)))
       => ( ! [X4: A,Xs2: list(A),Y2: B,Ys3: list(B)] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),P,X4),Y2))
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs2),Ys3))
               => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,Xs2),Ys3))
                 => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys3))) ) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,Xs),Ys)) ) ) ) ).

% list_all2_induct
tff(fact_7471_list_Orel__induct,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X: list(A),Y: list(B),Q: fun(list(A),fun(list(B),bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X),Y))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,nil(A)),nil(B)))
       => ( ! [A21: A,A222: list(A),B21: B,B222: list(B)] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),R,A21),B21))
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,A222),B222))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A21),A222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B21),B222))) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,X),Y)) ) ) ) ).

% list.rel_induct
tff(fact_7472_list_Orel__cases,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A2: list(A),B2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),A2),B2))
     => ( ( ( A2 = nil(A) )
         => ( B2 != nil(B) ) )
       => ~ ! [X15: A,X23: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),X23) )
             => ! [Y15: B,Y23: list(B)] :
                  ( ( B2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y15),Y23) )
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),R,X15),Y15))
                   => ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X23),Y23)) ) ) ) ) ) ).

% list.rel_cases
tff(fact_7473_list_Orel__distinct_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),Y21: B,Y222: list(B)] : ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y222))) ).

% list.rel_distinct(1)
tff(fact_7474_list_Orel__distinct_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),Y21: A,Y222: list(A)] : ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y222)),nil(B))) ).

% list.rel_distinct(2)
tff(fact_7475_list__all2__dropI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),N: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
     => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),drop(B),N),Ys))) ) ).

% list_all2_dropI
tff(fact_7476_list__all2__takeI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),N: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
     => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),N),Xs)),aa(list(B),list(B),aa(nat,fun(list(B),list(B)),take(B),N),Ys))) ) ).

% list_all2_takeI
tff(fact_7477_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ~ ? [F7: fun(nat,A)] :
          ! [I4: nat] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F7,aa(nat,nat,suc,I4))),aa(nat,A,F7,I4))),R2)) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_7478_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),F2: fun(nat,A)] :
      ( wf(A,R2)
     => ~ ! [K2: nat] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F2,aa(nat,nat,suc,K2))),aa(nat,A,F2,K2))),R2)) ) ).

% wf_no_infinite_down_chainE
tff(fact_7479_wf__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [P5: fun(A,bool)] :
          ( ! [X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R2))
                 => pp(aa(A,bool,P5,Y4)) )
             => pp(aa(A,bool,P5,X3)) )
         => ! [X_13: A] : pp(aa(A,bool,P5,X_13)) ) ) ).

% wf_def
tff(fact_7480_wfE__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Q: set(A)] :
      ( wf(A,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),Q))
       => ~ ! [Z3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),Q))
             => ~ ! [Y3: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R))
                   => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),Q)) ) ) ) ) ).

% wfE_min
tff(fact_7481_wfI__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X4: A,Q8: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Q8))
         => ? [Xa: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),Q8))
              & ! [Y2: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa)),R))
                 => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),Q8)) ) ) )
     => wf(A,R) ) ).

% wfI_min
tff(fact_7482_wfUNIVI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [P6: fun(A,bool),X4: A] :
          ( ! [Xa: A] :
              ( ! [Y2: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa)),R2))
                 => pp(aa(A,bool,P6,Y2)) )
             => pp(aa(A,bool,P6,Xa)) )
         => pp(aa(A,bool,P6,X4)) )
     => wf(A,R2) ) ).

% wfUNIVI
tff(fact_7483_wf__asym,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,X: A] :
      ( wf(A,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),X)),R2))
       => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A2)),R2)) ) ) ).

% wf_asym
tff(fact_7484_wf__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,bool),A2: A] :
      ( wf(A,R2)
     => ( ! [X4: A] :
            ( ! [Y3: A] :
                ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4)),R2))
               => pp(aa(A,bool,P,Y3)) )
           => pp(aa(A,bool,P,X4)) )
       => pp(aa(A,bool,P,A2)) ) ) ).

% wf_induct
tff(fact_7485_wf__irrefl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] :
      ( wf(A,R2)
     => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2)),R2)) ) ).

% wf_irrefl
tff(fact_7486_wf__not__sym,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,X: A] :
      ( wf(A,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),X)),R2))
       => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A2)),R2)) ) ) ).

% wf_not_sym
tff(fact_7487_wf__not__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] :
      ( wf(A,R2)
     => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2)),R2)) ) ).

% wf_not_refl
tff(fact_7488_wf__eq__minimal,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [Q7: set(A)] :
          ( ? [X3: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Q7))
         => ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Q7))
              & ! [Y4: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R2))
                 => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),Q7)) ) ) ) ) ).

% wf_eq_minimal
tff(fact_7489_wf__induct__rule,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,bool),A2: A] :
      ( wf(A,R2)
     => ( ! [X4: A] :
            ( ! [Y3: A] :
                ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4)),R2))
               => pp(aa(A,bool,P,Y3)) )
           => pp(aa(A,bool,P,X4)) )
       => pp(aa(A,bool,P,A2)) ) ) ).

% wf_induct_rule
tff(fact_7490_wfE__min_H,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Q: set(A)] :
      ( wf(A,R)
     => ( ( Q != bot_bot(set(A)) )
       => ~ ! [Z3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),Q))
             => ~ ! [Y3: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R))
                   => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),Q)) ) ) ) ) ).

% wfE_min'
tff(fact_7491_list__all2__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(A,fun(B,bool)),Us: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Vs)))
      <=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Us),Vs)) ) ) ) ).

% list_all2_append
tff(fact_7492_list__all2__append1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(A),Zs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Zs))
    <=> ? [Us2: list(B),Vs2: list(B)] :
          ( ( Zs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs2) )
          & ( aa(list(B),nat,size_size(list(B)),Us2) = aa(list(A),nat,size_size(list(A)),Xs) )
          & ( aa(list(B),nat,size_size(list(B)),Vs2) = aa(list(A),nat,size_size(list(A)),Ys) )
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Us2))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Ys),Vs2)) ) ) ).

% list_all2_append1
tff(fact_7493_list__all2__append2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),Zs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)))
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Vs2) )
          & ( aa(list(A),nat,size_size(list(A)),Us2) = aa(list(B),nat,size_size(list(B)),Ys) )
          & ( aa(list(A),nat,size_size(list(A)),Vs2) = aa(list(B),nat,size_size(list(B)),Zs) )
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Us2),Ys))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Vs2),Zs)) ) ) ).

% list_all2_append2
tff(fact_7494_list__all2__same,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),P),Xs),Xs))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P,X3),X3)) ) ) ).

% list_all2_same
tff(fact_7495_list_Orel__refl__strong,axiom,
    ! [A: $tType,X: list(A),Ra2: fun(A,fun(A,bool))] :
      ( ! [Z3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(list(A),set(A),set2(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),Ra2,Z3),Z3)) )
     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),list_all2(A,A),Ra2),X),X)) ) ).

% list.rel_refl_strong
tff(fact_7496_list_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X: list(A),Y: list(B),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X),Y))
     => ( ! [Z3: A,Yb: B] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(list(A),set(A),set2(A),X)))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Yb),aa(list(B),set(B),set2(B),Y)))
             => ( pp(aa(B,bool,aa(A,fun(B,bool),R,Z3),Yb))
               => pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z3),Yb)) ) ) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),Ra2),X),Y)) ) ) ).

% list.rel_mono_strong
tff(fact_7497_list_Orel__cong,axiom,
    ! [A: $tType,B: $tType,X: list(A),Ya: list(A),Y: list(B),Xa2: list(B),R: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( ( X = Ya )
     => ( ( Y = Xa2 )
       => ( ! [Z3: A,Yb: B] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),aa(list(A),set(A),set2(A),Ya)))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Yb),aa(list(B),set(B),set2(B),Xa2)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),R,Z3),Yb))
                <=> pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z3),Yb)) ) ) )
         => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),X),Y))
          <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),Ra2),Ya),Xa2)) ) ) ) ) ).

% list.rel_cong
tff(fact_7498_list__all2__lengthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% list_all2_lengthD
tff(fact_7499_list__all2__rev1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),aa(list(A),list(A),rev(A),Xs)),Ys))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),aa(list(B),list(B),rev(B),Ys))) ) ).

% list_all2_rev1
tff(fact_7500_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P2: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P2),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P2)),aa(nat,B,nth(B,Ys),P2))) ) ) ).

% list_all2_nthD
tff(fact_7501_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P2: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P2),aa(list(B),nat,size_size(list(B)),Ys)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P2)),aa(nat,B,nth(B,Ys),P2))) ) ) ).

% list_all2_nthD2
tff(fact_7502_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),A2)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,A2),N2)),aa(nat,B,nth(B,B2),N2))) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),A2),B2)) ) ) ).

% list_all2_all_nthI
tff(fact_7503_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,B,nth(B,Ys),I4))) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_7504_list_Orel__sel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A2: list(A),B2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),A2),B2))
    <=> ( ( ( A2 = nil(A) )
        <=> ( B2 = nil(B) ) )
        & ( ( A2 != nil(A) )
         => ( ( B2 != nil(B) )
           => ( pp(aa(B,bool,aa(A,fun(B,bool),R,aa(list(A),A,hd(A),A2)),aa(list(B),B,hd(B),B2)))
              & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(list(A),list(A),tl(A),A2)),aa(list(B),list(B),tl(B),B2))) ) ) ) ) ) ).

% list.rel_sel
tff(fact_7505_wfI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),B3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_agj(set(A),fun(A,set(A)),B3))))
     => ( ! [X4: A,P6: fun(A,bool)] :
            ( ! [Xa: A] :
                ( ! [Y2: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Xa)),R2))
                   => pp(aa(A,bool,P6,Y2)) )
               => pp(aa(A,bool,P6,Xa)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3))
               => pp(aa(A,bool,P6,X4)) ) ) )
       => wf(A,R2) ) ) ).

% wfI
tff(fact_7506_wf__eq__minimal2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [A8: set(A)] :
          ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),field2(A,R2)))
            & ( A8 != bot_bot(set(A)) ) )
         => ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A8))
              & ! [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A8))
                 => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3)),R2)) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_7507_product__lists__set,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),set(list(A)),set2(list(A)),aa(list(list(A)),list(list(A)),product_lists(A),Xss)) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_aja(list(list(A)),fun(list(A),bool),Xss)) ).

% product_lists_set
tff(fact_7508_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,set(B)),F2: fun(A,set(B))] :
      ( ! [A4: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4)),R2))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),Ub,A4)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Ub,B4)),aa(A,set(B),Ub,A4)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F2,B4)),aa(A,set(B),Ub,A4)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(A,set(B),F2,A4)),aa(A,set(B),F2,B4))) ) )
     => wf(A,R2) ) ).

% wf_bounded_set
tff(fact_7509_list__all2I,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
      ( ! [X4: product_prod(A,B)] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,A2,B2))))
         => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P),X4)) )
     => ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),A2),B2)) ) ) ).

% list_all2I
tff(fact_7510_finite__subset__wf,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_ajb(set(A),fun(set(A),fun(set(A),bool)),A3)))) ) ).

% finite_subset_wf
tff(fact_7511_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),P),Xs),Ys))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X3: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P),X3)) ) ) ) ).

% list_all2_iff
tff(fact_7512_dependent__wf__choice,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),P: fun(fun(A,B),fun(A,fun(B,bool)))] :
      ( wf(A,R)
     => ( ! [F3: fun(A,B),G3: fun(A,B),X4: A,R3: B] :
            ( ! [Z4: A] :
                ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X4)),R))
               => ( aa(A,B,F3,Z4) = aa(A,B,G3,Z4) ) )
           => ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X4),R3))
            <=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,G3),X4),R3)) ) )
       => ( ! [X4: A,F3: fun(A,B)] :
              ( ! [Y3: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4)),R))
                 => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),Y3),aa(A,B,F3,Y3))) )
             => ? [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X4),X_12)) )
         => ? [F3: fun(A,B)] :
            ! [X2: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X2),aa(A,B,F3,X2))) ) ) ) ).

% dependent_wf_choice
tff(fact_7513_partial__order__on__well__order__on,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( order_7125193373082350890der_on(A,A3,R2)
       => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id(A))) ) ) ).

% partial_order_on_well_order_on
tff(fact_7514_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_7515_dependent__wellorder__choice,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(A)
     => ! [P: fun(fun(A,B),fun(A,fun(B,bool)))] :
          ( ! [R3: B,F3: fun(A,B),G3: fun(A,B),X4: A] :
              ( ! [Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X4))
                 => ( aa(A,B,F3,Y3) = aa(A,B,G3,Y3) ) )
             => ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X4),R3))
              <=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,G3),X4),R3)) ) )
         => ( ! [X4: A,F3: fun(A,B)] :
                ( ! [Y3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X4))
                   => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),Y3),aa(A,B,F3,Y3))) )
               => ? [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X4),X_12)) )
           => ? [F3: fun(A,B)] :
              ! [X2: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X2),aa(A,B,F3,X2))) ) ) ) ).

% dependent_wellorder_choice
tff(fact_7516_finite__Partial__order__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,P: fun(A,bool)] :
      ( order_7125193373082350890der_on(A,field2(A,R2),R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),field2(A,R2)))
       => ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),field2(A,R2)))
               => ( ! [Y3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),order_aboveS(A,R2,X4)))
                     => pp(aa(A,bool,P,Y3)) )
                 => pp(aa(A,bool,P,X4)) ) )
           => pp(aa(A,bool,P,X)) ) ) ) ) ).

% finite_Partial_order_induct
tff(fact_7517_chains__extend,axiom,
    ! [A: $tType,C2: set(set(A)),S2: set(set(A)),Z: set(A)] :
      ( pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),C2),chains2(A,S2)))
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Z),S2))
       => ( ! [X4: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),C2))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Z)) )
         => pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Z),bot_bot(set(set(A))))),C2)),chains2(A,S2))) ) ) ) ).

% chains_extend
tff(fact_7518_Zorns__po__lemma,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,field2(A,R2),R2)
     => ( ! [C7: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C7),chains(A,R2)))
           => ? [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),field2(A,R2)))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),C7))
                   => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X2)),R2)) ) ) )
       => ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),field2(A,R2)))
            & ! [Xa: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),field2(A,R2)))
               => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),R2))
                 => ( Xa = X4 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_7519_Zorn__Lemma2,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ! [X4: set(set(A))] :
          ( pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),X4),chains2(A,A3)))
         => ? [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa),A3))
              & ! [Xb3: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xb3),X4))
                 => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xb3),Xa)) ) ) )
     => ? [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),A3))
          & ! [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa),A3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Xa))
               => ( Xa = X4 ) ) ) ) ) ).

% Zorn_Lemma2
tff(fact_7520_chainsD,axiom,
    ! [A: $tType,C2: set(set(A)),S2: set(set(A)),X: set(A),Y: set(A)] :
      ( pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),C2),chains2(A,S2)))
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X),C2))
       => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y),C2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X),Y))
            | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Y),X)) ) ) ) ) ).

% chainsD
tff(fact_7521_Zorn__Lemma,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ! [X4: set(set(A))] :
          ( pp(aa(set(set(set(A))),bool,aa(set(set(A)),fun(set(set(set(A))),bool),member(set(set(A))),X4),chains2(A,A3)))
         => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X4)),A3)) )
     => ? [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),A3))
          & ! [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa),A3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Xa))
               => ( Xa = X4 ) ) ) ) ) ).

% Zorn_Lemma
tff(fact_7522_Chains__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : chains(A,R2) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ajc(set(product_prod(A,A)),fun(set(A),bool),R2)) ).

% Chains_def
tff(fact_7523_Chains__subset_H,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R2)
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_wl(set(product_prod(A,A)),fun(A,fun(A,bool)),R2)))),chains(A,R2))) ) ).

% Chains_subset'
tff(fact_7524_power_Opower__eq__if,axiom,
    ! [A: $tType,M: nat,One: A,Times: fun(A,fun(A,A)),P2: A] :
      ( ( ( M = zero_zero(nat) )
       => ( power2(A,One,Times,P2,M) = One ) )
      & ( ( M != zero_zero(nat) )
       => ( power2(A,One,Times,P2,M) = aa(A,A,aa(A,fun(A,A),Times,P2),power2(A,One,Times,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ).

% power.power_eq_if
tff(fact_7525_chains__alt__def,axiom,
    ! [A: $tType,A3: set(set(A))] : chains2(A,A3) = aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),pred_chain(set(A),A3,ord_less(set(A)))) ).

% chains_alt_def
tff(fact_7526_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A2: A] : power2(A,One,Times,A2,zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_7527_subset__Zorn,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ! [C7: set(set(A))] :
          ( pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),C7))
         => ? [X2: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X2),A3))
              & ! [Xa4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa4),C7))
                 => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa4),X2)) ) ) )
     => ? [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),A3))
          & ! [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa),A3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Xa))
               => ( Xa = X4 ) ) ) ) ) ).

% subset_Zorn
tff(fact_7528_subset_Ochain__def,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),C3))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C3),A3))
        & ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),C3))
           => ! [Xa3: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa3),C3))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X3),Xa3))
                  | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Xa3),X3)) ) ) ) ) ) ).

% subset.chain_def
tff(fact_7529_subset_OchainI,axiom,
    ! [A: $tType,C3: set(set(A)),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C3),A3))
     => ( ! [X4: set(A),Y2: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),C3))
           => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y2),C3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X4),Y2))
                | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Y2),X4)) ) ) )
       => pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),C3)) ) ) ).

% subset.chainI
tff(fact_7530_pred__on_OchainI,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),P: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),A3))
     => ( ! [X4: A,Y2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),C3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),C3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),X4),Y2))
                | pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),Y2),X4)) ) ) )
       => pp(aa(set(A),bool,pred_chain(A,A3,P),C3)) ) ) ).

% pred_on.chainI
tff(fact_7531_pred__on_Ochain__def,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,fun(A,bool)),C3: set(A)] :
      ( pp(aa(set(A),bool,pred_chain(A,A3,P),C3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),A3))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),C3))
           => ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),C3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),X3),Xa3))
                  | pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),Xa3),X3)) ) ) ) ) ) ).

% pred_on.chain_def
tff(fact_7532_subset__Zorn_H,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ! [C7: set(set(A))] :
          ( pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),C7))
         => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C7)),A3)) )
     => ? [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),A3))
          & ! [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa),A3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Xa))
               => ( Xa = X4 ) ) ) ) ) ).

% subset_Zorn'
tff(fact_7533_subset_Ochain__empty,axiom,
    ! [A: $tType,A3: set(set(A))] : pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),bot_bot(set(set(A))))) ).

% subset.chain_empty
tff(fact_7534_pred__on_Ochain__empty,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,fun(A,bool))] : pp(aa(set(A),bool,pred_chain(A,A3,P),bot_bot(set(A)))) ).

% pred_on.chain_empty
tff(fact_7535_subset_Ochain__total,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(set(A)),X: set(A),Y: set(A)] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),C3))
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X),C3))
       => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y),C3))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X),Y))
            | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Y),X)) ) ) ) ) ).

% subset.chain_total
tff(fact_7536_subset__chain__def,axiom,
    ! [A: $tType,A19: set(set(A)),C10: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A19,ord_less(set(A))),C10))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C10),A19))
        & ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),C10))
           => ! [Xa3: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa3),C10))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa3))
                  | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa3),X3)) ) ) ) ) ) ).

% subset_chain_def
tff(fact_7537_subset__chain__insert,axiom,
    ! [A: $tType,A19: set(set(A)),B3: set(A),B11: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A19,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),B3),B11)))
    <=> ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B3),A19))
        & ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),B11))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),B3))
              | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),X3)) ) )
        & pp(aa(set(set(A)),bool,pred_chain(set(A),A19,ord_less(set(A))),B11)) ) ) ).

% subset_chain_insert
tff(fact_7538_subset__Zorn__nonempty,axiom,
    ! [A: $tType,A19: set(set(A))] :
      ( ( A19 != bot_bot(set(set(A))) )
     => ( ! [C11: set(set(A))] :
            ( ( C11 != bot_bot(set(set(A))) )
           => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A19,ord_less(set(A))),C11))
             => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C11)),A19)) ) )
       => ? [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),A19))
            & ! [Xa: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa),A19))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Xa))
                 => ( Xa = X4 ) ) ) ) ) ) ).

% subset_Zorn_nonempty
tff(fact_7539_Union__in__chain,axiom,
    ! [A: $tType,B11: set(set(A)),A19: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),B11))
     => ( ( B11 != bot_bot(set(set(A))) )
       => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A19,ord_less(set(A))),B11))
         => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11)),B11)) ) ) ) ).

% Union_in_chain
tff(fact_7540_Inter__in__chain,axiom,
    ! [A: $tType,B11: set(set(A)),A19: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),B11))
     => ( ( B11 != bot_bot(set(set(A))) )
       => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A19,ord_less(set(A))),B11))
         => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)),B11)) ) ) ) ).

% Inter_in_chain
tff(fact_7541_subset_Ochain__extend,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(set(A)),Z: set(A)] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),C3))
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Z),A3))
       => ( ! [X4: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),C3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X4),Z)) )
         => pp(aa(set(set(A)),bool,pred_chain(set(A),A3,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))))),C3))) ) ) ) ).

% subset.chain_extend
tff(fact_7542_pred__on_Ochain__extend,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,fun(A,bool)),C3: set(A),Z: A] :
      ( pp(aa(set(A),bool,pred_chain(A,A3,P),C3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),A3))
       => ( ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P),fequal(A)),X4),Z)) )
         => pp(aa(set(A),bool,pred_chain(A,A3,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)))),C3))) ) ) ) ).

% pred_on.chain_extend
tff(fact_7543_finite__subset__Union__chain,axiom,
    ! [A: $tType,A3: set(A),B11: set(set(A)),A19: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11)))
       => ( ( B11 != bot_bot(set(set(A))) )
         => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A19,ord_less(set(A))),B11))
           => ~ ! [B8: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B8),B11))
                 => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B8)) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_7544_Chains__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),chains(A,R2)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_wl(set(product_prod(A,A)),fun(A,fun(A,bool)),R2))))) ).

% Chains_subset
tff(fact_7545_Chains__alt__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R2)
     => ( chains(A,R2) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_wl(set(product_prod(A,A)),fun(A,fun(A,bool)),R2))) ) ) ).

% Chains_alt_def
tff(fact_7546_chain__subset__alt__def,axiom,
    ! [A: $tType,C3: set(set(A))] :
      ( chain_subset(A,C3)
    <=> pp(aa(set(set(A)),bool,pred_chain(set(A),top_top(set(set(A))),ord_less(set(A))),C3)) ) ).

% chain_subset_alt_def
tff(fact_7547_cauchyD,axiom,
    ! [X6: fun(nat,rat),R2: rat] :
      ( cauchy(X6)
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R2))
       => ? [K2: nat] :
          ! [M2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M2))
           => ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N5))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M2)),aa(nat,rat,X6,N5)))),R2)) ) ) ) ) ).

% cauchyD
tff(fact_7548_chain__subset__def,axiom,
    ! [A: $tType,C3: set(set(A))] :
      ( chain_subset(A,C3)
    <=> ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),C3))
         => ! [Xa3: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Xa3),C3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa3))
                | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa3),X3)) ) ) ) ) ).

% chain_subset_def
tff(fact_7549_cauchy__imp__bounded,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ? [B4: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B4))
          & ! [N5: nat] : pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N5))),B4)) ) ) ).

% cauchy_imp_bounded
tff(fact_7550_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B4: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B4))
            & ? [K2: nat] :
                ( ! [N5: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N5))
                   => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B4),aa(rat,rat,uminus_uminus(rat),aa(nat,rat,X6,N5)))) )
                | ! [N5: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N5))
                   => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B4),aa(nat,rat,X6,N5))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_7551_cauchy__not__vanishes,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B4: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B4))
            & ? [K2: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N5))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B4),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N5)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_7552_cauchy__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
    <=> ! [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
         => ? [K3: nat] :
            ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),M3))
             => ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N3))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M3)),aa(nat,rat,X6,N3)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_7553_cauchyI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3))
         => ? [K8: nat] :
            ! [M5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K8),M5))
             => ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K8),N2))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M5)),aa(nat,rat,X6,N2)))),R3)) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_7554_le__Real,axiom,
    ! [X6: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y5)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real2(X6)),real2(Y5)))
        <=> ! [R5: rat] :
              ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
             => ? [K3: nat] :
                ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N3))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(nat,rat,X6,N3)),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Y5,N3)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_7555_finite__def,axiom,
    ! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),bool),aTP_Lamp_yk(fun(set(A),bool),fun(set(A),bool))) ).

% finite_def
tff(fact_7556_lfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),X: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F4))
         => ( ( aa(A,A,F4,X) = X )
           => ( ! [Z3: A] :
                  ( ( aa(A,A,F4,Z3) = Z3 )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z3)) )
             => ( complete_lattice_lfp(A,F4) = X ) ) ) ) ) ).

% lfp_eqI
tff(fact_7557_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,fun(A,A))] :
          ( ! [X4: A,Y2: A,W: A,Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X4),W)),aa(A,A,aa(A,fun(A,A),F2,Y2),Z3))) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_ajd(fun(A,fun(A,A)),fun(A,A),F2)) = complete_lattice_lfp(A,aTP_Lamp_aje(fun(A,fun(A,A)),fun(A,A),F2)) ) ) ) ).

% lfp_lfp
tff(fact_7558_lfp__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),A3: A] :
          ( ! [U3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,U3)),U3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),U3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),complete_lattice_lfp(A,F2))) ) ) ).

% lfp_greatest
tff(fact_7559_lfp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,A3)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F2)),A3)) ) ) ).

% lfp_lowerbound
tff(fact_7560_lfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),G: fun(A,A)] :
          ( ! [Z8: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,Z8)),aa(A,A,G,Z8)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F2)),complete_lattice_lfp(A,G))) ) ) ).

% lfp_mono
tff(fact_7561_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,bool),set(A),collect(A),aTP_Lamp_ajf(fun(A,A),fun(A,bool),F2))) ) ).

% lfp_def
tff(fact_7562_def__lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,F2: fun(A,A),P: A] :
          ( ( A3 = complete_lattice_lfp(A,F2) )
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),P))),P))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),P)) ) ) ) ) ).

% def_lfp_induct
tff(fact_7563_lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F2)),P))),P))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F2)),P)) ) ) ) ).

% lfp_induct
tff(fact_7564_lfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: fun(A,bool)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( ! [S6: A] :
                ( pp(aa(A,bool,P,S6))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),S6),complete_lattice_lfp(A,F2)))
                 => pp(aa(A,bool,P,aa(A,A,F2,S6))) ) )
           => ( ! [M8: set(A)] :
                  ( ! [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),M8))
                     => pp(aa(A,bool,P,X2)) )
                 => pp(aa(A,bool,P,aa(set(A),A,complete_Sup_Sup(A),M8))) )
             => pp(aa(A,bool,P,complete_lattice_lfp(A,F2))) ) ) ) ) ).

% lfp_ordinal_induct
tff(fact_7565_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F2)) = complete_lattice_lfp(A,F2) ) ) ) ).

% lfp_funpow
tff(fact_7566_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( pp(aa(fun(A,A),bool,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_7567_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X2: fun(A,A)] : comple7512665784863727008ratesp(A,X2) = complete_lattice_lfp(fun(A,bool),aTP_Lamp_zx(fun(A,A),fun(fun(A,bool),fun(A,bool)),X2)) ) ).

% iteratesp_def
tff(fact_7568_not__positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ pp(aa(real,bool,positive2,real2(X6)))
      <=> ! [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
           => ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N3))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(nat,rat,X6,N3)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_7569_less__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
    <=> pp(aa(real,bool,positive2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),X))) ) ).

% less_real_def
tff(fact_7570_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,F2: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),complete_lattice_lfp(set(product_prod(A,B)),F2)))
     => ( pp(aa(fun(set(product_prod(A,B)),set(product_prod(A,B))),bool,order_mono(set(product_prod(A,B)),set(product_prod(A,B))),F2))
       => ( ! [A4: A,B4: B] :
              ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),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),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P))))))
             => pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B4)) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,A2),B2)) ) ) ) ).

% lfp_induct2
tff(fact_7571_iteratesp_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),A2: A] :
          ( pp(aa(A,bool,comple7512665784863727008ratesp(A,F2),A2))
        <=> ( ? [X3: A] :
                ( ( A2 = aa(A,A,F2,X3) )
                & pp(aa(A,bool,comple7512665784863727008ratesp(A,F2),X3)) )
            | ? [M9: set(A)] :
                ( ( A2 = aa(set(A),A,complete_Sup_Sup(A),M9) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M9)
                & ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),M9))
                   => pp(aa(A,bool,comple7512665784863727008ratesp(A,F2),X3)) ) ) ) ) ) ).

% iteratesp.simps
tff(fact_7572_iteratesp_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),A2: A] :
          ( pp(aa(A,bool,comple7512665784863727008ratesp(A,F2),A2))
         => ( ! [X4: A] :
                ( ( A2 = aa(A,A,F2,X4) )
               => ~ pp(aa(A,bool,comple7512665784863727008ratesp(A,F2),X4)) )
           => ~ ! [M8: set(A)] :
                  ( ( A2 = aa(set(A),A,complete_Sup_Sup(A),M8) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                   => ~ ! [X2: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),M8))
                         => pp(aa(A,bool,comple7512665784863727008ratesp(A,F2),X2)) ) ) ) ) ) ) ).

% iteratesp.cases
tff(fact_7573_iteratesp_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M4: set(A),F2: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M4)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),M4))
               => pp(aa(A,bool,comple7512665784863727008ratesp(A,F2),X4)) )
           => pp(aa(A,bool,comple7512665784863727008ratesp(A,F2),aa(set(A),A,complete_Sup_Sup(A),M4))) ) ) ) ).

% iteratesp.Sup
tff(fact_7574_positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( pp(aa(real,bool,positive2,real2(X6)))
      <=> ? [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
            & ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N3))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,X6,N3))) ) ) ) ) ).

% positive_Real
tff(fact_7575_lfp__transfer__bounded,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [P: fun(A,bool),F2: fun(A,A),Alpha: fun(A,B),G: fun(B,B)] :
          ( pp(aa(A,bool,P,bot_bot(A)))
         => ( ! [X4: A] :
                ( pp(aa(A,bool,P,X4))
               => pp(aa(A,bool,P,aa(A,A,F2,X4))) )
           => ( ! [M8: fun(nat,A)] :
                  ( ! [I3: nat] : pp(aa(A,bool,P,aa(nat,A,M8,I3)))
                 => pp(aa(A,bool,P,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,M8),top_top(set(nat)))))) )
             => ( ! [M8: fun(nat,A)] :
                    ( pp(aa(fun(nat,A),bool,order_mono(nat,A),M8))
                   => ( ! [I3: nat] : pp(aa(A,bool,P,aa(nat,A,M8,I3)))
                     => ( aa(A,B,Alpha,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,M8),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_ajg(fun(A,B),fun(fun(nat,A),fun(nat,B)),Alpha),M8)),top_top(set(nat)))) ) ) )
               => ( order_sup_continuous(A,A,F2)
                 => ( order_sup_continuous(B,B,G)
                   => ( ! [X4: A] :
                          ( pp(aa(A,bool,P,X4))
                         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),complete_lattice_lfp(A,F2)))
                           => ( aa(A,B,Alpha,aa(A,A,F2,X4)) = aa(B,B,G,aa(A,B,Alpha,X4)) ) ) )
                     => ( ! [X4: B] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G,X4)))
                       => ( aa(A,B,Alpha,complete_lattice_lfp(A,F2)) = complete_lattice_lfp(B,G) ) ) ) ) ) ) ) ) ) ) ).

% lfp_transfer_bounded
tff(fact_7576_sup__continuous__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A)] :
          ( order_sup_continuous(A,A,F4)
         => ( complete_lattice_lfp(A,F4) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_ajh(fun(A,A),fun(nat,A),F4)),top_top(set(nat)))) ) ) ) ).

% sup_continuous_lfp
tff(fact_7577_lfp__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [Alpha: fun(A,B),F2: fun(A,A),G: fun(B,B)] :
          ( order_sup_continuous(A,B,Alpha)
         => ( order_sup_continuous(A,A,F2)
           => ( order_sup_continuous(B,B,G)
             => ( ! [X4: B] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G,X4)))
               => ( ! [X4: A] :
                      ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),complete_lattice_lfp(A,F2)))
                     => ( aa(A,B,Alpha,aa(A,A,F2,X4)) = aa(B,B,G,aa(A,B,Alpha,X4)) ) )
                 => ( aa(A,B,Alpha,complete_lattice_lfp(A,F2)) = complete_lattice_lfp(B,G) ) ) ) ) ) ) ) ).

% lfp_transfer
tff(fact_7578_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)
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F2))
           => ( ( aa(A,B,Alpha,bot_bot(A)) = bot_bot(B) )
             => ( ! [X4: A] : aa(A,B,Alpha,aa(A,A,F2,X4)) = aa(B,B,G,aa(A,B,Alpha,X4))
               => ( aa(A,B,Alpha,order_532582986084564980_cclfp(A,F2)) = order_532582986084564980_cclfp(B,G) ) ) ) ) ) ) ).

% cclfp_transfer
tff(fact_7579_Real_Opositive_Orep__eq,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,positive2,X))
    <=> ? [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
          & ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N3))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),rep_real(X,N3))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_7580_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_adb(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ) ).

% ord_class.lexordp_def
tff(fact_7581_lexordp__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ys: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),nil(A)),Ys))
        <=> ( Ys != nil(A) ) ) ) ).

% lexordp_simps(1)
tff(fact_7582_lexordp__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] : ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),nil(A))) ) ).

% lexordp_simps(2)
tff(fact_7583_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A),Y: A,Ys: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
              & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys)) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_7584_lexordp__append__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ys: list(A),Xs: list(A)] :
          ( ( Ys != nil(A) )
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))) ) ) ).

% lexordp_append_rightI
tff(fact_7585_lexordp__append__leftI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Us: list(A),Vs: list(A),Xs: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us),Vs))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs))) ) ) ).

% lexordp_append_leftI
tff(fact_7586_lexordp__append__leftD,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A),Us: list(A),Vs: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs)))
         => ( ! [A4: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A4),A4))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us),Vs)) ) ) ) ).

% lexordp_append_leftD
tff(fact_7587_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A,Xs: list(A),Ys: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ) ).

% lexordp.Cons
tff(fact_7588_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A,Xs: list(A),Ys: list(A)] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
           => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_7589_lexordp__irreflexive,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X4))
         => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Xs)) ) ) ).

% lexordp_irreflexive
tff(fact_7590_lexordp__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
         => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Ys),Xs)) ) ) ).

% lexordp_antisym
tff(fact_7591_lexordp__trans,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A),Zs: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
         => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Ys),Zs))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Zs)) ) ) ) ).

% lexordp_trans
tff(fact_7592_lexordp__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
          | ( Xs = Ys )
          | pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Ys),Xs)) ) ) ).

% lexordp_linear
tff(fact_7593_lexordp__irreflexive_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xs: list(A)] : ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Xs)) ) ).

% lexordp_irreflexive'
tff(fact_7594_lexordp_ONil,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Y: A,Ys: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ).

% lexordp.Nil
tff(fact_7595_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A),P: fun(list(A),fun(list(A),bool))] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
         => ( ! [Y2: A,Ys3: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)))
           => ( ! [X4: A,Xs2: list(A),Y2: A,Ys3: list(A)] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2))
                 => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3))) )
             => ( ! [X4: A,Xs2: list(A),Ys3: list(A)] :
                    ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys3))
                   => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),Ys3))
                     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3))) ) )
               => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs),Ys)) ) ) ) ) ) ).

% lexordp_induct
tff(fact_7596_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
         => ( ( ( Xs = nil(A) )
             => ! [Y2: A,Ys5: list(A)] : Ys != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5) )
           => ( ! [X4: A] :
                  ( ? [Xs4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4)
                 => ! [Y2: A] :
                      ( ? [Ys5: list(A)] : Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys5)
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2)) ) )
             => ~ ! [X4: A,Xs4: list(A)] :
                    ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
                   => ! [Ys5: list(A)] :
                        ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys5) )
                       => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs4),Ys5)) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_7597_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
            | ? [X3: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4)) )
            | ? [X3: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X3))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs3),Ys4)) ) ) ) ) ).

% lexordp.simps
tff(fact_7598_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
         => ( ( ( A1 = nil(A) )
             => ! [Y2: A,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
           => ( ! [X4: A] :
                  ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)
                 => ! [Y2: A] :
                      ( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2)) ) )
             => ~ ! [X4: A,Y2: A,Xs2: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
                   => ! [Ys3: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
                       => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y2))
                         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X4))
                           => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys3)) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_7599_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
        <=> ( ? [X3: A,Vs2: list(A)] : Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Vs2))
            | ? [Us2: list(A),A7: A,B7: A,Vs2: list(A),Ws3: list(A)] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A7),B7))
                & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),Vs2)) )
                & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B7),Ws3)) ) ) ) ) ) ).

% lexordp_iff
tff(fact_7600_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A,Us: list(A),Xs: list(A),Ys: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))) ) ) ).

% lexordp_append_left_rightI
tff(fact_7601_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
        <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),ord_less(A)))))) ) ) ).

% lexordp_conv_lexord
tff(fact_7602_ord_Olexordp__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool))] : lexordp2(A,Less) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_ada(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Less)) ).

% ord.lexordp_def
tff(fact_7603_finite__refines__card__le,axiom,
    ! [A: $tType,A3: set(A),R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A3,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),S2))
       => ( equiv_equiv(A,A3,R)
         => ( equiv_equiv(A,A3,S2)
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A3,S2))),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A3,R)))) ) ) ) ) ).

% finite_refines_card_le
tff(fact_7604_ord_Olexordp__simps_I3_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Xs: list(A),Y: A,Ys: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))
    <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
        | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y),X))
          & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),Ys)) ) ) ) ).

% ord.lexordp_simps(3)
tff(fact_7605_ord_Olexordp__simps_I2_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A)] : ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),nil(A))) ).

% ord.lexordp_simps(2)
tff(fact_7606_ord_Olexordp__simps_I1_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),Ys: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),nil(A)),Ys))
    <=> ( Ys != nil(A) ) ) ).

% ord.lexordp_simps(1)
tff(fact_7607_ord_Olexordp_Osimps,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),A1: list(A),A22: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),A1),A22))
    <=> ( ? [Y4: A,Ys4: list(A)] :
            ( ( A1 = nil(A) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
        | ? [X3: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & pp(aa(A,bool,aa(A,fun(A,bool),Less,X3),Y4)) )
        | ? [X3: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X3),Y4))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y4),X3))
            & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs3),Ys4)) ) ) ) ).

% ord.lexordp.simps
tff(fact_7608_ord_Olexordp_Ocases,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),A1: list(A),A22: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),A1),A22))
     => ( ( ( A1 = nil(A) )
         => ! [Y2: A,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
       => ( ! [X4: A] :
              ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2)
             => ! [Y2: A] :
                  ( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3)
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X4),Y2)) ) )
         => ~ ! [X4: A,Y2: A,Xs2: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
               => ! [Ys3: list(A)] :
                    ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys3) )
                   => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X4),Y2))
                     => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y2),X4))
                       => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs2),Ys3)) ) ) ) ) ) ) ) ).

% ord.lexordp.cases
tff(fact_7609_ord_Olexordp_ONil,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),Y: A,Ys: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ).

% ord.lexordp.Nil
tff(fact_7610_ord_Olexordp__irreflexive,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A)] :
      ( ! [X4: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X4),X4))
     => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),Xs)) ) ).

% ord.lexordp_irreflexive
tff(fact_7611_ord_Olexordp_Ocong,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool))] : lexordp2(A,Less) = lexordp2(A,Less) ).

% ord.lexordp.cong
tff(fact_7612_ord_Olexordp_OCons__eq,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A),Ys: list(A)] :
      ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
     => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Y),X))
       => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),Ys))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ) ) ).

% ord.lexordp.Cons_eq
tff(fact_7613_ord_Olexordp_OCons,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A),Ys: list(A)] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) ).

% ord.lexordp.Cons
tff(fact_7614_ord_Olexordp__append__leftD,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),Xs: list(A),Us: list(A),Vs: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs)))
     => ( ! [A4: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A4),A4))
       => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Us),Vs)) ) ) ).

% ord.lexordp_append_leftD
tff(fact_7615_ord_Olexordp__append__leftI,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),Us: list(A),Vs: list(A),Xs: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Us),Vs))
     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs))) ) ).

% ord.lexordp_append_leftI
tff(fact_7616_ord_Olexordp__append__rightI,axiom,
    ! [A: $tType,Ys: list(A),Less: fun(A,fun(A,bool)),Xs: list(A)] :
      ( ( Ys != nil(A) )
     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))) ) ).

% ord.lexordp_append_rightI
tff(fact_7617_ord_Olexordp__append__left__rightI,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),X: A,Y: A,Us: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),Less,X),Y))
     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),lexordp2(A,Less),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))) ) ).

% ord.lexordp_append_left_rightI
tff(fact_7618_equiv__listrel,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( equiv_equiv(A,A3,R2)
     => equiv_equiv(list(A),lists(A,A3),listrel(A,A,R2)) ) ).

% equiv_listrel
tff(fact_7619_in__quotient__imp__non__empty,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
       => ( X6 != bot_bot(set(A)) ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_7620_in__quotient__imp__subset,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),A3)) ) ) ).

% in_quotient_imp_subset
tff(fact_7621_quotient__eqI,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X6: set(A),Y5: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
       => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y5),equiv_quotient(A,A3,R2)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),Y5))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
               => ( X6 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_7622_quotient__eq__iff,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X6: set(A),Y5: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
       => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y5),equiv_quotient(A,A3,R2)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),Y5))
             => ( ( X6 = Y5 )
              <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_7623_in__quotient__imp__closed,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),X6)) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_7624_quotient__disj,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X6: set(A),Y5: set(A)] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
       => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y5),equiv_quotient(A,A3,R2)))
         => ( ( X6 = Y5 )
            | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X6),Y5) = bot_bot(set(A)) ) ) ) ) ) ).

% quotient_disj
tff(fact_7625_finite__refines__finite,axiom,
    ! [A: $tType,A3: set(A),R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A3,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),S2))
       => ( equiv_equiv(A,A3,R)
         => ( equiv_equiv(A,A3,S2)
           => pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A3,S2))) ) ) ) ) ).

% finite_refines_finite
tff(fact_7626_eq__equiv__class__iff2,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
         => ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))),R2) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))),R2) )
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_7627_equiv__imp__dvd__card,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( equiv_equiv(A,A3,R2)
       => ( ! [X7: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X7),equiv_quotient(A,A3,R2)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),X7))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% equiv_imp_dvd_card
tff(fact_7628_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),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),Y),bot_bot(set(A))))),X6))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_7629_UN__equiv__class__inject,axiom,
    ! [B: $tType,A: $tType,A3: set(A),R2: set(product_prod(A,A)),F2: fun(A,set(B)),X6: set(A),Y5: set(A)] :
      ( equiv_equiv(A,A3,R2)
     => ( equiv_congruent(A,set(B),R2,F2)
       => ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),X6)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),Y5)) )
         => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
           => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y5),equiv_quotient(A,A3,R2)))
             => ( ! [X4: A,Y2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                   => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
                     => ( ( aa(A,set(B),F2,X4) = aa(A,set(B),F2,Y2) )
                       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y2)),R2)) ) ) )
               => ( X6 = Y5 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
tff(fact_7630_proj__iff,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),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),Y),bot_bot(set(A))))),A3))
       => ( ( equiv_proj(A,A,R2,X) = equiv_proj(A,A,R2,Y) )
        <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)) ) ) ) ).

% proj_iff
tff(fact_7631_congruentI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F2: fun(A,B)] :
      ( ! [Y2: A,Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2))
         => ( aa(A,B,F2,Y2) = aa(A,B,F2,Z3) ) )
     => equiv_congruent(A,B,R2,F2) ) ).

% congruentI
tff(fact_7632_congruentD,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F2: fun(A,B),Y: A,Z: A] :
      ( equiv_congruent(A,B,R2,F2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),R2))
       => ( aa(A,B,F2,Y) = aa(A,B,F2,Z) ) ) ) ).

% congruentD
tff(fact_7633_UN__equiv__class,axiom,
    ! [B: $tType,A: $tType,A3: set(A),R2: set(product_prod(A,A)),F2: fun(A,set(B)),A2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( equiv_congruent(A,set(B),R2,F2)
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
         => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) = aa(A,set(B),F2,A2) ) ) ) ) ).

% UN_equiv_class
tff(fact_7634_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),N: nat,S: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),set_ord_lessThan(A,S)))))
           => ( aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),set_ord_lessThan(A,S))),N) = aa(nat,A,infini527867602293511546merate(A,S2),N) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_7635_ImageI,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,R2: set(product_prod(A,B)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),R2))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),image(A,B,R2,A3))) ) ) ).

% ImageI
tff(fact_7636_Image__empty2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(B,A))] : image(B,A,R,bot_bot(set(B))) = bot_bot(set(A)) ).

% Image_empty2
tff(fact_7637_Image__empty1,axiom,
    ! [B: $tType,A: $tType,X6: set(B)] : image(B,A,bot_bot(set(product_prod(B,A))),X6) = bot_bot(set(A)) ).

% Image_empty1
tff(fact_7638_Image__singleton__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A2: B] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A2),B2)),R2)) ) ).

% Image_singleton_iff
tff(fact_7639_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),M: nat,N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),M)),aa(nat,A,infini527867602293511546merate(A,S2),N)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% enumerate_mono_iff
tff(fact_7640_listrel__Nil,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,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)),nil(B)),bot_bot(set(list(B))))) = 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)))) ).

% listrel_Nil
tff(fact_7641_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),M: nat,N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(set(A),nat,finite_card(A),S2)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),S2)))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),M)),aa(nat,A,infini527867602293511546merate(A,S2),N)))
              <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_7642_equiv__class__self,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ).

% equiv_class_self
tff(fact_7643_proj__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X: B] : equiv_proj(B,A,R2,X) = image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))) ).

% proj_def
tff(fact_7644_countable__Image,axiom,
    ! [B: $tType,A: $tType,Y5: set(A),X6: set(product_prod(A,B))] :
      ( ! [Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),Y5))
         => countable_countable(B,image(A,B,X6,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y2),bot_bot(set(A))))) )
     => ( countable_countable(A,Y5)
       => countable_countable(B,image(A,B,X6,Y5)) ) ) ).

% countable_Image
tff(fact_7645_finite__rtrancl__Image,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R))
     => ( pp(aa(set(A),bool,finite_finite2(A),A3))
       => pp(aa(set(A),bool,finite_finite2(A),image(A,A,transitive_rtrancl(A,R),A3))) ) ) ).

% finite_rtrancl_Image
tff(fact_7646_strict__mono__enumerate,axiom,
    ! [S2: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => order_strict_mono(nat,nat,infini527867602293511546merate(nat,S2)) ) ).

% strict_mono_enumerate
tff(fact_7647_enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,infini527867602293511546merate(A,S2),N)),S2)) ) ) ).

% enumerate_in_set
tff(fact_7648_enumerate__Ex,axiom,
    ! [S2: set(nat),S: nat] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),S),S2))
       => ? [N2: nat] : aa(nat,nat,infini527867602293511546merate(nat,S2),N2) = S ) ) ).

% enumerate_Ex
tff(fact_7649_finite__Image,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R))
     => pp(aa(set(B),bool,finite_finite2(B),image(A,B,R,A3))) ) ).

% finite_Image
tff(fact_7650_quotientE,axiom,
    ! [A: $tType,X6: set(A),A3: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A3,R2)))
     => ~ ! [X4: A] :
            ( ( X6 = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X4),bot_bot(set(A)))) )
           => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3)) ) ) ).

% quotientE
tff(fact_7651_quotientI,axiom,
    ! [A: $tType,X: A,A3: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),equiv_quotient(A,A3,R2))) ) ).

% quotientI
tff(fact_7652_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A2: B] : image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B)))) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_aji(set(product_prod(B,A)),fun(B,fun(A,bool)),R2),A2)) ).

% Image_singleton
tff(fact_7653_Image__mono,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,B)),R2: set(product_prod(A,B)),A11: set(A),A3: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R4),R2))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),A3))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,R4,A11)),image(A,B,R2,A3))) ) ) ).

% Image_mono
tff(fact_7654_Image__closed__trancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R2,X6)),X6))
     => ( image(A,A,transitive_rtrancl(A,R2),X6) = X6 ) ) ).

% Image_closed_trancl
tff(fact_7655_le__enumerate,axiom,
    ! [S2: set(nat),N: nat] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,infini527867602293511546merate(nat,S2),N))) ) ).

% le_enumerate
tff(fact_7656_rev__ImageI,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B2: B,R2: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),R2))
       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),image(A,B,R2,A3))) ) ) ).

% rev_ImageI
tff(fact_7657_Image__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),image(B,A,R2,A3)))
    <=> ? [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
          & pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),B2)),R2)) ) ) ).

% Image_iff
tff(fact_7658_ImageE,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),image(B,A,R2,A3)))
     => ~ ! [X4: B] :
            ( pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),B2)),R2))
           => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3)) ) ) ).

% ImageE
tff(fact_7659_Image__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(A)] : image(A,B,R2,S) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_ajj(set(product_prod(A,B)),fun(set(A),fun(B,bool)),R2),S)) ).

% Image_def
tff(fact_7660_Image__Int__subset,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A3: set(B),B3: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(B,A,R,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(B,A,R,A3)),image(B,A,R,B3)))) ).

% Image_Int_subset
tff(fact_7661_wfI__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),image(A,A,R,A5)))
         => ( A5 = bot_bot(set(A)) ) )
     => wf(A,R) ) ).

% wfI_pf
tff(fact_7662_wfE__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: set(A)] :
      ( wf(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),image(A,A,R,A3)))
       => ( A3 = bot_bot(set(A)) ) ) ) ).

% wfE_pf
tff(fact_7663_Image__INT__subset,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),B3: fun(C,set(B)),A3: set(C)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(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),B3),A3)))),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_ajk(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B3)),A3)))) ).

% Image_INT_subset
tff(fact_7664_Image__subset,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),A3: set(A),B3: set(B),C3: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,R2,C3)),B3)) ) ).

% Image_subset
tff(fact_7665_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),N)),aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,N)))) ) ) ).

% enumerate_step
tff(fact_7666_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,N: nat,S2: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),M)),aa(nat,A,infini527867602293511546merate(A,S2),N))) ) ) ) ).

% enumerate_mono
tff(fact_7667_listrel__Cons,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X: B,Xs: list(B)] : 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),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),bot_bot(set(list(B))))) = set_Cons(A,image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),X),bot_bot(set(B)))),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_7668_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y5: set(A)] :
          ( ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X6)))
             => ( aa(nat,A,infini527867602293511546merate(A,X6),I2) = aa(nat,A,infini527867602293511546merate(A,Y5),I2) ) )
         => ( pp(aa(set(A),bool,finite_finite2(A),X6))
           => ( pp(aa(set(A),bool,finite_finite2(A),Y5))
             => ( ( aa(set(A),nat,finite_card(A),X6) = aa(set(A),nat,finite_card(A),Y5) )
               => ( X6 = Y5 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_7669_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),S: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S),S2))
           => ? [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(set(A),nat,finite_card(A),S2)))
                & ( aa(nat,A,infini527867602293511546merate(A,S2),N2) = S ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_7670_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),S2)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,infini527867602293511546merate(A,S2),N)),S2)) ) ) ) ).

% finite_enumerate_in_set
tff(fact_7671_eq__equiv__class,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A,A3: set(A)] :
      ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
     => ( equiv_equiv(A,A3,R2)
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) ) ) ) ).

% eq_equiv_class
tff(fact_7672_equiv__class__eq,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
       => ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_7673_eq__equiv__class__iff,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
         => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_7674_equiv__class__eq__iff,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
      <=> ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3)) ) ) ) ).

% equiv_class_eq_iff
tff(fact_7675_refines__equiv__class__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A)),A3: set(A),A2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),S2))
     => ( equiv_equiv(A,A3,R)
       => ( equiv_equiv(A,A3,S2)
         => ( image(A,A,R,image(A,A,S2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = image(A,A,S2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_7676_refines__equiv__class__eq2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A)),A3: set(A),A2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),S2))
     => ( equiv_equiv(A,A3,R)
       => ( equiv_equiv(A,A3,S2)
         => ( image(A,A,S2,image(A,A,R,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = image(A,A,S2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_7677_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( order_7125193373082350890der_on(A,field2(A,R2),R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
         => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_7678_Image__eq__UN,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),B3: set(B)] : image(B,A,R2,B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ajl(set(product_prod(B,A)),fun(B,set(A)),R2)),B3)) ).

% Image_eq_UN
tff(fact_7679_range__enumerate,axiom,
    ! [S2: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => ( aa(set(nat),set(nat),image2(nat,nat,infini527867602293511546merate(nat,S2)),top_top(set(nat))) = S2 ) ) ).

% range_enumerate
tff(fact_7680_inj__enumerate,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
         => inj_on(nat,A,infini527867602293511546merate(A,S2),top_top(set(nat))) ) ) ).

% inj_enumerate
tff(fact_7681_equiv__class__subset,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))) ) ) ).

% equiv_class_subset
tff(fact_7682_subset__equiv__class,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),B2: A,A2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) ) ) ) ).

% subset_equiv_class
tff(fact_7683_equiv__class__nondisjoint,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X: A,A2: A,B2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) ) ) ).

% equiv_class_nondisjoint
tff(fact_7684_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,N: nat,S2: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( pp(aa(set(A),bool,finite_finite2(A),S2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),S2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),M)),aa(nat,A,infini527867602293511546merate(A,S2),N))) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_7685_finite__le__enumerate,axiom,
    ! [S2: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(nat),nat,finite_card(nat),S2)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,infini527867602293511546merate(nat,S2),N))) ) ) ).

% finite_le_enumerate
tff(fact_7686_bij__enumerate,axiom,
    ! [S2: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S2))
     => bij_betw(nat,nat,infini527867602293511546merate(nat,S2),top_top(set(nat)),S2) ) ).

% bij_enumerate
tff(fact_7687_singleton__quotient,axiom,
    ! [A: $tType,X: A,R2: set(product_prod(A,A))] : equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),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)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% singleton_quotient
tff(fact_7688_finite__bij__enumerate,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => bij_betw(nat,A,infini527867602293511546merate(A,S2),set_ord_lessThan(nat,aa(set(A),nat,finite_card(A),S2)),S2) ) ) ).

% finite_bij_enumerate
tff(fact_7689_quotient__def,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] : equiv_quotient(A,A3,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_ajm(set(product_prod(A,A)),fun(A,set(set(A))),R2)),A3)) ).

% quotient_def
tff(fact_7690_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),N)),aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,N)))) ) ) ) ).

% finite_enumerate_step
tff(fact_7691_Image__fold,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R))
     => ( image(A,B,R,S2) = 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_ajn(set(A),fun(A,fun(B,fun(set(B),set(B)))),S2)),bot_bot(set(B)),R) ) ) ).

% Image_fold
tff(fact_7692_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),N: nat] : aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,N)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,infini527867602293511546merate(A,S2),zero_zero(nat))),bot_bot(set(A))))),N) ) ).

% enumerate_Suc'
tff(fact_7693_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y5: set(A)] :
          ( ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X6)))
             => ( aa(nat,A,infini527867602293511546merate(A,X6),I2) = aa(nat,A,infini527867602293511546merate(A,Y5),I2) ) )
         => ( pp(aa(set(A),bool,finite_finite2(A),X6))
           => ( pp(aa(set(A),bool,finite_finite2(A),Y5))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X6)),aa(set(A),nat,finite_card(A),Y5)))
               => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),Y5)) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_7694_congruent2__implies__congruent__UN,axiom,
    ! [A: $tType,C: $tType,B: $tType,A15: set(A),R1: set(product_prod(A,A)),A25: set(B),R22: set(product_prod(B,B)),F2: fun(A,fun(B,set(C))),A2: B] :
      ( equiv_equiv(A,A15,R1)
     => ( equiv_equiv(B,A25,R22)
       => ( equiv_congruent2(A,B,set(C),R1,R22,F2)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A25))
           => equiv_congruent(A,set(C),R1,aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_ajo(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F2),A2)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_7695_UN__equiv__class2,axiom,
    ! [A: $tType,C: $tType,B: $tType,A15: set(A),R1: set(product_prod(A,A)),A25: set(B),R22: set(product_prod(B,B)),F2: fun(A,fun(B,set(C))),A1: A,A22: B] :
      ( equiv_equiv(A,A15,R1)
     => ( equiv_equiv(B,A25,R22)
       => ( equiv_congruent2(A,B,set(C),R1,R22,F2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A1),A15))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A22),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_ajo(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F2),A22)),image(A,A,R1,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F2,A1),A22) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_7696_congruent2I_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F2: fun(A,fun(B,C))] :
      ( ! [Y15: A,Z1: A,Y23: B,Z22: B] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y15),Z1)),R1))
         => ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y23),Z22)),R22))
           => ( aa(B,C,aa(A,fun(B,C),F2,Y15),Y23) = aa(B,C,aa(A,fun(B,C),F2,Z1),Z22) ) ) )
     => equiv_congruent2(A,B,C,R1,R22,F2) ) ).

% congruent2I'
tff(fact_7697_congruent2D,axiom,
    ! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F2: fun(A,fun(B,C)),Y1: A,Z12: A,Y22: B,Z23: B] :
      ( equiv_congruent2(A,B,C,R1,R22,F2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z12)),R1))
       => ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y22),Z23)),R22))
         => ( aa(B,C,aa(A,fun(B,C),F2,Y1),Y22) = aa(B,C,aa(A,fun(B,C),F2,Z12),Z23) ) ) ) ) ).

% congruent2D
tff(fact_7698_congruent2I,axiom,
    ! [C: $tType,B: $tType,A: $tType,A15: set(A),R1: set(product_prod(A,A)),A25: set(B),R22: set(product_prod(B,B)),F2: fun(A,fun(B,C))] :
      ( equiv_equiv(A,A15,R1)
     => ( equiv_equiv(B,A25,R22)
       => ( ! [Y2: A,Z3: A,W: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),W),A25))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R1))
               => ( aa(B,C,aa(A,fun(B,C),F2,Y2),W) = aa(B,C,aa(A,fun(B,C),F2,Z3),W) ) ) )
         => ( ! [Y2: B,Z3: B,W: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),W),A15))
               => ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z3)),R22))
                 => ( aa(B,C,aa(A,fun(B,C),F2,W),Y2) = aa(B,C,aa(A,fun(B,C),F2,W),Z3) ) ) )
           => equiv_congruent2(A,B,C,R1,R22,F2) ) ) ) ) ).

% congruent2I
tff(fact_7699_congruent2__commuteI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),R2: set(product_prod(A,A)),F2: fun(A,fun(A,B))] :
      ( equiv_equiv(A,A3,R2)
     => ( ! [Y2: A,Z3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),A3))
             => ( aa(A,B,aa(A,fun(A,B),F2,Y2),Z3) = aa(A,B,aa(A,fun(A,B),F2,Z3),Y2) ) ) )
       => ( ! [Y2: A,Z3: A,W: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),W),A3))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z3)),R2))
               => ( aa(A,B,aa(A,fun(A,B),F2,W),Y2) = aa(A,B,aa(A,fun(A,B),F2,W),Z3) ) ) )
         => equiv_congruent2(A,A,B,R2,R2,F2) ) ) ) ).

% congruent2_commuteI
tff(fact_7700_subset__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( order_preorder_on(A,field2(A,R2),R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))))
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A2)),R2)) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_7701_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S2)))
           => ( aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_ajp(set(A),fun(nat,fun(A,bool)),S2),N)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_7702_Least__eq__0,axiom,
    ! [P: fun(nat,bool)] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ord_Least(nat,P) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_7703_Least__Suc,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_ajq(fun(nat,bool),fun(nat,bool),P))) ) ) ) ).

% Least_Suc
tff(fact_7704_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),ord_Least(A,P)))
         => ~ pp(aa(A,bool,P,K)) ) ) ).

% not_less_Least
tff(fact_7705_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),K: A] :
          ( pp(aa(A,bool,P,K))
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% LeastI
tff(fact_7706_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
         => ( ! [X4: A] :
                ( pp(aa(A,bool,P,X4))
               => pp(aa(A,bool,Q,X4)) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_ex
tff(fact_7707_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool)] :
          ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% LeastI_ex
tff(fact_7708_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A2: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,A2))
         => ( ! [X4: A] :
                ( pp(aa(A,bool,P,X4))
               => pp(aa(A,bool,Q,X4)) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2
tff(fact_7709_Least__Suc2,axiom,
    ! [P: fun(nat,bool),N: nat,Q: fun(nat,bool),M: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( pp(aa(nat,bool,Q,M))
       => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
         => ( ! [K2: nat] :
                ( pp(aa(nat,bool,P,aa(nat,nat,suc,K2)))
              <=> pp(aa(nat,bool,Q,K2)) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_7710_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),K: A] :
          ( pp(aa(A,bool,P,K))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),K)) ) ) ).

% Least_le
tff(fact_7711_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
         => ( ! [A4: A] :
                ( pp(aa(A,bool,P,A4))
               => ( ! [B9: A] :
                      ( pp(aa(A,bool,P,B9))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),B9)) )
                 => pp(aa(A,bool,Q,A4)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_7712_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A2: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,A2))
         => ( ! [A4: A] :
                ( pp(aa(A,bool,P,A4))
               => ( ! [B9: A] :
                      ( pp(aa(A,bool,P,B9))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),B9)) )
                 => pp(aa(A,bool,Q,A4)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).

% LeastI2_wellorder
tff(fact_7713_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y2: A] :
                ( pp(aa(A,bool,P,Y2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) )
           => ( ord_Least(A,P) = X ) ) ) ) ).

% Least_equality
tff(fact_7714_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y2: A] :
                ( pp(aa(A,bool,P,Y2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y2)) )
           => ( ! [X4: A] :
                  ( pp(aa(A,bool,P,X4))
                 => ( ! [Y3: A] :
                        ( pp(aa(A,bool,P,Y3))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y3)) )
                   => pp(aa(A,bool,Q,X4)) ) )
             => pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ) ).

% LeastI2_order
tff(fact_7715_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),Z: A] :
          ( ? [X2: A] :
              ( pp(aa(A,bool,P,X2))
              & ! [Y2: A] :
                  ( pp(aa(A,bool,P,Y2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2)) )
              & ! [Y2: A] :
                  ( ( pp(aa(A,bool,P,Y2))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Ya2)) ) )
                 => ( Y2 = X2 ) ) )
         => ( pp(aa(A,bool,P,Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),Z)) ) ) ) ).

% Least1_le
tff(fact_7716_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] :
          ( ? [X2: A] :
              ( pp(aa(A,bool,P,X2))
              & ! [Y2: A] :
                  ( pp(aa(A,bool,P,Y2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2)) )
              & ! [Y2: A] :
                  ( ( pp(aa(A,bool,P,Y2))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y2),Ya2)) ) )
                 => ( Y2 = X2 ) ) )
         => pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).

% Least1I
tff(fact_7717_Least__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P)))
         => ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
           => ( ord_Least(A,P) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,bool),set(A),collect(A),P)) ) ) ) ) ).

% Least_Min
tff(fact_7718_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] : aa(nat,A,infini527867602293511546merate(A,S2),zero_zero(nat)) = ord_Least(A,aTP_Lamp_ajr(set(A),fun(A,bool),S2)) ) ).

% enumerate_0
tff(fact_7719_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S2: set(A),P: fun(A,bool)] : abort_Bleast(A,S2,P) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ajs(set(A),fun(fun(A,bool),fun(A,bool)),S2),P)) ) ).

% abort_Bleast_def
tff(fact_7720_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S2: set(A),P: fun(A,bool)] : bleast(A,S2,P) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ajs(set(A),fun(fun(A,bool),fun(A,bool)),S2),P)) ) ).

% Bleast_def
tff(fact_7721_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_7722_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F2))
         => ( ? [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S2))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa4)) ) )
           => ( ord_Least(B,aa(set(A),fun(B,bool),aTP_Lamp_ajt(fun(A,B),fun(set(A),fun(B,bool)),F2),S2)) = aa(A,B,F2,ord_Least(A,aTP_Lamp_aju(set(A),fun(A,bool),S2))) ) ) ) ) ).

% Least_mono
tff(fact_7723_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,bool)] : ord_Least(A,P) = the(A,aTP_Lamp_ajv(fun(A,bool),fun(A,bool),P)) ) ).

% Least_def
tff(fact_7724_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_ajp(set(A),fun(nat,fun(A,bool)),S2),N)) ) ) ) ).

% enumerate_Suc''
tff(fact_7725_subset__Image__Image__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),B3: set(A)] :
      ( order_preorder_on(A,field2(A,R2),R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R2)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R2,A3)),image(A,A,R2,B3)))
          <=> ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ? [Xa3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),B3))
                    & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3)),R2)) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_7726_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),N: nat] : aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,N)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),ord_Least(A,aTP_Lamp_ajr(set(A),fun(A,bool),S2))),bot_bot(set(A))))),N) ) ).

% enumerate_Suc
tff(fact_7727_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( refl_on(A,field2(A,R2),R2)
     => ( antisym(A,R2)
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
           => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
            <=> ( A2 = B2 ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_7728_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,N)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7729_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% mask_nat_positive_iff
tff(fact_7730_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( ( bit_se2239418461657761734s_mask(A,N) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_7731_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_7732_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_7733_antisym__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisym(A,R2)
    <=> ! [X3: A,Y4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4)),R2))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R2))
           => ( X3 = Y4 ) ) ) ) ).

% antisym_def
tff(fact_7734_antisymI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y2)),R2))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X4)),R2))
           => ( X4 = Y2 ) ) )
     => antisym(A,R2) ) ).

% antisymI
tff(fact_7735_antisymD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( antisym(A,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
       => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A2)),R2))
         => ( A2 = B2 ) ) ) ) ).

% antisymD
tff(fact_7736_less__eq__mask,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),bit_se2239418461657761734s_mask(nat,N))) ).

% less_eq_mask
tff(fact_7737_antisym__empty,axiom,
    ! [A: $tType] : antisym(A,bot_bot(set(product_prod(A,A)))) ).

% antisym_empty
tff(fact_7738_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2239418461657761734s_mask(int,N)),zero_zero(int))) ).

% not_mask_negative_int
tff(fact_7739_antisym__singleton,axiom,
    ! [A: $tType,X: 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)),X),bot_bot(set(product_prod(A,A))))) ).

% antisym_singleton
tff(fact_7740_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),bit_se2239418461657761734s_mask(nat,N))) ) ).

% less_mask
tff(fact_7741_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),bit_se2239418461657761734s_mask(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% mask_nat_less_exp
tff(fact_7742_lexn_Osimps_I2_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),aa(nat,nat,suc,N)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),inf_inf(set(product_prod(list(A),list(A)))),aa(set(product_prod(product_prod(A,list(A)),product_prod(A,list(A)))),set(product_prod(list(A),list(A))),image2(product_prod(product_prod(A,list(A)),product_prod(A,list(A))),product_prod(list(A),list(A)),product_map_prod(product_prod(A,list(A)),list(A),product_prod(A,list(A)),list(A),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)))),lex_prod(A,list(A),R2,aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),N)))),aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_ajw(nat,fun(list(A),fun(list(A),bool)),N)))) ).

% lexn.simps(2)
tff(fact_7743_stable__sort__key__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Sk: fun(fun(B,A),fun(list(B),list(B)))] :
          ( linord3483353639454293061rt_key(B,A,Sk)
        <=> ! [F7: fun(B,A),Xs3: list(B),K3: A] : aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(A,fun(B,bool),aTP_Lamp_ajx(fun(B,A),fun(A,fun(B,bool)),F7),K3)),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),Sk,F7),Xs3)) = aa(list(B),list(B),aa(fun(B,bool),fun(list(B),list(B)),filter2(B),aa(A,fun(B,bool),aTP_Lamp_ajx(fun(B,A),fun(A,fun(B,bool)),F7),K3)),Xs3) ) ) ).

% stable_sort_key_def
tff(fact_7744_map__prod__simp,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(C,A),G: fun(D,B),A2: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F2,G),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A2),B2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,A2)),aa(D,B,G,B2)) ).

% map_prod_simp
tff(fact_7745_map__prod__imageI,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R: set(product_prod(A,B)),F2: fun(A,C),G: fun(B,D)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),R))
     => pp(aa(set(product_prod(C,D)),bool,aa(product_prod(C,D),fun(set(product_prod(C,D)),bool),member(product_prod(C,D)),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,F2,A2)),aa(B,D,G,B2))),aa(set(product_prod(A,B)),set(product_prod(C,D)),image2(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F2,G)),R))) ) ).

% map_prod_imageI
tff(fact_7746_prod__fun__imageE,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod(A,B),F2: fun(C,A),G: fun(D,B),R: set(product_prod(C,D))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),C2),aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F2,G)),R)))
     => ~ ! [X4: C,Y2: D] :
            ( ( C2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,X4)),aa(D,B,G,Y2)) )
           => ~ pp(aa(set(product_prod(C,D)),bool,aa(product_prod(C,D),fun(set(product_prod(C,D)),bool),member(product_prod(C,D)),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X4),Y2)),R)) ) ) ).

% prod_fun_imageE
tff(fact_7747_map__prod__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F2: fun(A,C),G: fun(B,D)] : product_map_prod(A,C,B,D,F2,G) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_ajy(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F2),G)) ).

% map_prod_def
tff(fact_7748_stable__sort__key__sort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => linord3483353639454293061rt_key(A,B,linorder_sort_key(A,B)) ) ).

% stable_sort_key_sort_key
tff(fact_7749_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb2: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb2,Xc) = Y )
     => ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb2),Xc))))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa2))
               => ( Y = Xc ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa2))
               => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb2,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc)) ) ) )
           => ~ accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb2),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_7750_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2))))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A2))
         => ( set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc2) = Acc2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A2))
         => ( set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc2) = set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A2),Acc2)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_7751_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))))] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A1),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A22),A32))))
     => ( ! [F3: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc: A] :
            ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B4),Acc))))
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B4),A4))
               => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F3,A4),Acc))) )
             => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F3),A4),B4),Acc)) ) )
       => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,A0),A1),A22),A32)) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_7752_divmod__nat__code,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa(product_prod(code_integer,code_integer),product_prod(nat,nat),product_map_prod(code_integer,nat,code_integer,nat,code_nat_of_integer,code_nat_of_integer),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),code_integer_of_nat(M)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),code_integer_of_nat(N)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),code_integer_of_nat(M)),code_divmod_abs(code_integer_of_nat(M),code_integer_of_nat(N))))) ).

% divmod_nat_code
tff(fact_7753_Gcd__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite2(A),A3))
           => ( semiring_gcd_Gcd_fin(A,A3) = finite_fold(A,A,gcd_gcd(A),zero_zero(A),A3) ) )
          & ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
           => ( semiring_gcd_Gcd_fin(A,A3) = one_one(A) ) ) ) ) ).

% Gcd_fin.eq_fold
tff(fact_7754_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_7755_Gcd__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( semiring_gcd_Gcd_fin(A,A3) = one_one(A) ) ) ) ).

% Gcd_fin.infinite
tff(fact_7756_Gcd__fin__eq__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( semiring_gcd_Gcd_fin(A,A3) = gcd_Gcd(A,A3) ) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_7757_dvd__gcd__list__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,Xs: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Xs))))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),X3)) ) ) ) ).

% dvd_gcd_list_iff
tff(fact_7758_gcd__list__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Bs: list(A),A2: A] :
          ( ! [B4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),aa(list(A),set(A),set2(A),Bs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B4)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Bs)))) ) ) ).

% gcd_list_greatest
tff(fact_7759_dvd__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),B2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),semiring_gcd_Gcd_fin(A,A3)))
          <=> ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),X3)) ) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_7760_Gcd__fin__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),B4)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A2),semiring_gcd_Gcd_fin(A,A3))) ) ) ) ).

% Gcd_fin_greatest
tff(fact_7761_Gcd__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B3: set(A),A3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),semiring_gcd_Gcd_fin(A,B3)),semiring_gcd_Gcd_fin(A,A3)) = semiring_gcd_Gcd_fin(A,A3) ) ) ) ).

% Gcd_fin.subset
tff(fact_7762_integer__of__nat__0,axiom,
    code_integer_of_nat(zero_zero(nat)) = zero_zero(code_integer) ).

% integer_of_nat_0
tff(fact_7763_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
         => ( semiring_gcd_Gcd_fin(A,A3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_7764_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] : semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ).

% Gcd_fin.insert_remove
tff(fact_7765_Gcd__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Xs)) = aa(A,A,aa(list(A),fun(A,A),aa(fun(A,fun(A,A)),fun(list(A),fun(A,A)),fold(A,A),gcd_gcd(A)),Xs),zero_zero(A)) ) ).

% Gcd_fin.set_eq_fold
tff(fact_7766_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ( semiring_gcd_Gcd_fin(A,A3) = zero_zero(A) )
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A)))))
            & pp(aa(set(A),bool,finite_finite2(A),A3)) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_7767_independent__explicit__finite__subsets,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( ~ real_V358717886546972837endent(A,A3)
        <=> ! [S10: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S10),A3))
             => ( pp(aa(set(A),bool,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_ajz(fun(A,real),fun(A,A),U6)),S10) = zero_zero(A) )
                   => ! [X3: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S10))
                       => ( aa(A,real,U6,X3) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_finite_subsets
tff(fact_7768_independent__explicit__module,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( ~ real_V358717886546972837endent(A,S)
        <=> ! [T4: set(A),U6: fun(A,real),V6: A] :
              ( pp(aa(set(A),bool,finite_finite2(A),T4))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T4),S))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),U6)),T4) = zero_zero(A) )
                 => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V6),T4))
                   => ( aa(A,real,U6,V6) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_module
tff(fact_7769_independent__empty,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ~ real_V358717886546972837endent(A,bot_bot(set(A))) ) ).

% independent_empty
tff(fact_7770_dependent__single,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] :
          ( real_V358717886546972837endent(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))
        <=> ( X = zero_zero(A) ) ) ) ).

% dependent_single
tff(fact_7771_independent__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ~ real_V358717886546972837endent(A,A3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
           => ~ real_V358717886546972837endent(A,B3) ) ) ) ).

% independent_mono
tff(fact_7772_dependent__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),A3: set(A)] :
          ( real_V358717886546972837endent(A,B3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
           => real_V358717886546972837endent(A,A3) ) ) ) ).

% dependent_mono
tff(fact_7773_dependent__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A3))
         => real_V358717886546972837endent(A,A3) ) ) ).

% dependent_zero
tff(fact_7774_independent__Union__directed,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [C3: set(set(A))] :
          ( ! [C4: set(A),D3: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C4),C3))
             => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),D3),C3))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),D3))
                  | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D3),C4)) ) ) )
         => ( ! [C4: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C4),C3))
               => ~ real_V358717886546972837endent(A,C4) )
           => ~ real_V358717886546972837endent(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) ) ) ) ).

% independent_Union_directed
tff(fact_7775_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) )
               => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V3),Basis)) )
           => ( ! [V3: A] :
                  ( ( aa(A,real,G,V3) != zero_zero(real) )
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V3),Basis)) )
             => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),F2))))
               => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),G))))
                 => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),F2)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),F2))) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),G)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),G))) )
                   => ( F2 = G ) ) ) ) ) ) ) ) ).

% unique_representation
tff(fact_7776_independent__if__scalars__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A3))
         => ( ! [F3: fun(A,real),X4: A] :
                ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),F3)),A3) = zero_zero(A) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => ( aa(A,real,F3,X4) = zero_zero(real) ) ) )
           => ~ real_V358717886546972837endent(A,A3) ) ) ) ).

% independent_if_scalars_zero
tff(fact_7777_dependent__finite,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( real_V358717886546972837endent(A,S2)
          <=> ? [U6: fun(A,real)] :
                ( ? [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                    & ( aa(A,real,U6,X3) != zero_zero(real) ) )
                & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),U6)),S2) = zero_zero(A) ) ) ) ) ) ).

% dependent_finite
tff(fact_7778_independentD__unique,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),X6: fun(A,real),Y5: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,B3)
         => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X6))))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X6))),B3))
             => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),Y5))))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),Y5))),B3))
                 => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),X6)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X6))) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),Y5)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),Y5))) )
                   => ( X6 = Y5 ) ) ) ) ) ) ) ) ).

% independentD_unique
tff(fact_7779_independentD,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),T2: set(A),U: fun(A,real),V2: A] :
          ( ~ real_V358717886546972837endent(A,S)
         => ( pp(aa(set(A),bool,finite_finite2(A),T2))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),U)),T2) = zero_zero(A) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),T2))
                 => ( aa(A,real,U,V2) = zero_zero(real) ) ) ) ) ) ) ) ).

% independentD
tff(fact_7780_dependent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] :
          ( real_V358717886546972837endent(A,B3)
        <=> ? [X8: fun(A,real)] :
              ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X8))))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X8))),B3))
              & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),X8)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X8))) = zero_zero(A) )
              & ? [X3: A] : aa(A,real,X8,X3) != zero_zero(real) ) ) ) ).

% dependent_alt
tff(fact_7781_independent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] :
          ( ~ real_V358717886546972837endent(A,B3)
        <=> ! [X8: fun(A,real)] :
              ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X8))))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X8))),B3))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),X8)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X8))) = zero_zero(A) )
                 => ! [X3: A] : aa(A,real,X8,X3) = zero_zero(real) ) ) ) ) ) ).

% independent_alt
tff(fact_7782_independentD__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),X6: fun(A,real),X: A] :
          ( ~ real_V358717886546972837endent(A,B3)
         => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X6))))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X6))),B3))
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),X6)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),X6))) = zero_zero(A) )
               => ( aa(A,real,X6,X) = zero_zero(real) ) ) ) ) ) ) ).

% independentD_alt
tff(fact_7783_dependent__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( real_V358717886546972837endent(A,S)
        <=> ? [T4: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),T4))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T4),S))
              & ? [U6: fun(A,real)] :
                  ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),U6)),T4) = zero_zero(A) )
                  & ? [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),T4))
                      & ( aa(A,real,U6,X3) != zero_zero(real) ) ) ) ) ) ) ).

% dependent_explicit
tff(fact_7784_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_7785_relation__of__def,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),A3: set(A)] : order_relation_of(A,P,A3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_akb(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),P),A3))) ).

% relation_of_def
tff(fact_7786_Chains__relation__of,axiom,
    ! [A: $tType,C3: set(A),P: fun(A,fun(A,bool)),A3: set(A)] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C3),chains(A,order_relation_of(A,P,A3))))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C3),A3)) ) ).

% Chains_relation_of
tff(fact_7787_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_7788_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),N: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,N))
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_7789_map__add__find__right,axiom,
    ! [B: $tType,A: $tType,N: fun(B,option(A)),K: B,Xx: A,M: fun(B,option(A))] :
      ( ( aa(B,option(A),N,K) = aa(A,option(A),some(A),Xx) )
     => ( aa(B,option(A),map_add(B,A,M,N),K) = aa(A,option(A),some(A),Xx) ) ) ).

% map_add_find_right
tff(fact_7790_map__add__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),G: fun(A,option(B))] :
      ( ! [X3: A] : aa(A,option(B),map_add(A,B,F2,G),X3) = none(B)
    <=> ( ! [X3: A] : aa(A,option(B),F2,X3) = none(B)
        & ! [X3: A] : aa(A,option(B),G,X3) = none(B) ) ) ).

% map_add_eq_empty_iff
tff(fact_7791_map__add__None,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K: B] :
      ( ( aa(B,option(A),map_add(B,A,M,N),K) = none(A) )
    <=> ( ( aa(B,option(A),N,K) = none(A) )
        & ( aa(B,option(A),M,K) = none(A) ) ) ) ).

% map_add_None
tff(fact_7792_empty__eq__map__add__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( aTP_Lamp_aat(A,option(B)) = map_add(A,B,F2,G) )
    <=> ( ! [X3: A] : aa(A,option(B),F2,X3) = none(B)
        & ! [X3: A] : aa(A,option(B),G,X3) = none(B) ) ) ).

% empty_eq_map_add_iff
tff(fact_7793_map__add__empty,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,M,aTP_Lamp_aat(A,option(B))) = M ).

% map_add_empty
tff(fact_7794_empty__map__add,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,aTP_Lamp_aat(A,option(B)),M) = M ).

% empty_map_add
tff(fact_7795_map__add__upd,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),G: fun(A,option(B)),X: A,Y: B] : map_add(A,B,F2,fun_upd(A,option(B),G,X,aa(B,option(B),some(B),Y))) = fun_upd(A,option(B),map_add(A,B,F2,G),X,aa(B,option(B),some(B),Y)) ).

% map_add_upd
tff(fact_7796_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,I))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
           => pp(bit_se6407376104438227557le_bit(A,Tyrep,J)) ) ) ) ).

% possible_bit_less_imp
tff(fact_7797_map__add__Some__iff,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M,N),K) = aa(A,option(A),some(A),X) )
    <=> ( ( aa(B,option(A),N,K) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N,K) = none(A) )
          & ( aa(B,option(A),M,K) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_Some_iff
tff(fact_7798_map__add__SomeD,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M,N),K) = aa(A,option(A),some(A),X) )
     => ( ( aa(B,option(A),N,K) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N,K) = none(A) )
          & ( aa(B,option(A),M,K) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_SomeD
tff(fact_7799_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : pp(bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat))) ) ).

% possible_bit_0
tff(fact_7800_map__add__def,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X2: A] : aa(A,option(B),map_add(A,B,M1,M22),X2) = case_option(option(B),B,aa(A,option(B),M1,X2),some(B),aa(A,option(B),M22,X2)) ).

% map_add_def
tff(fact_7801_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_7802_map__add__upd__left,axiom,
    ! [A: $tType,B: $tType,M: A,E22: fun(A,option(B)),E1: fun(A,option(B)),U1: B] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),dom(A,B,E22)))
     => ( map_add(A,B,fun_upd(A,option(B),E1,M,aa(B,option(B),some(B),U1)),E22) = fun_upd(A,option(B),map_add(A,B,E1,E22),M,aa(B,option(B),some(B),U1)) ) ) ).

% map_add_upd_left
tff(fact_7803_map__add__map__of__foldr,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Ps: list(product_prod(A,B))] : map_add(A,B,M,map_of(A,B,Ps)) = aa(fun(A,option(B)),fun(A,option(B)),aa(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B))),aa(fun(product_prod(A,B),fun(fun(A,option(B)),fun(A,option(B)))),fun(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),foldr(product_prod(A,B),fun(A,option(B))),aa(fun(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),fun(product_prod(A,B),fun(fun(A,option(B)),fun(A,option(B)))),product_case_prod(A,B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_akc(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))))),Ps),M) ).

% map_add_map_of_foldr
tff(fact_7804_finite__range__map__of__map__add,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),L: list(product_prod(B,A))] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),F2),top_top(set(B)))))
     => pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),map_add(B,A,F2,map_of(B,A,L))),top_top(set(B))))) ) ).

% finite_range_map_of_map_add
tff(fact_7805_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_7806_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : bit_se4197421643247451524op_bit(A,M,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N),bit_se6407376104438227557le_bit(A,type2(A),N)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% drop_bit_exp_eq
tff(fact_7807_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% bit_minus_2_iff
tff(fact_7808_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_7809_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_7810_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,M)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% bit_mask_iff
tff(fact_7811_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X4: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X4) = zero_zero(A) )
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),C2),X4)) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_7812_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),semiri4206861660011772517g_char(A,type2(A))),N)) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_7813_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
           => ( aa(nat,A,semiring_1_of_nat(A),N3) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7814_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C2))
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X4))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),C2))
                   => ( aa(nat,A,semiring_1_of_nat(A),X4) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7815_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A))))
      <=> ? [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
            & ( aa(nat,A,semiring_1_of_nat(A),N3) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7816_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se4730199178511100633sh_bit(A,M,A2)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% bit_push_bit_iff
tff(fact_7817_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
        <=> ~ pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).

% fold_possible_bit
tff(fact_7818_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7819_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7820_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
            & ( N != zero_zero(nat) )
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ).

% bit_double_iff
tff(fact_7821_map__rec,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs) = aa(list(B),list(A),aa(fun(B,fun(list(B),fun(list(A),list(A)))),fun(list(B),list(A)),aa(list(A),fun(fun(B,fun(list(B),fun(list(A),list(A)))),fun(list(B),list(A))),rec_list(list(A),B),nil(A)),aTP_Lamp_akd(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),F2)),Xs) ).

% map_rec
tff(fact_7822_acyclic__insert,axiom,
    ! [A: $tType,Y: A,X: A,R2: set(product_prod(A,A))] :
      ( transitive_acyclic(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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R2))
    <=> ( transitive_acyclic(A,R2)
        & ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2))) ) ) ).

% acyclic_insert
tff(fact_7823_wf__iff__acyclic__if__finite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( wf(A,R2)
      <=> transitive_acyclic(A,R2) ) ) ).

% wf_iff_acyclic_if_finite
tff(fact_7824_finite__acyclic__wf,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( transitive_acyclic(A,R2)
       => wf(A,R2) ) ) ).

% finite_acyclic_wf
tff(fact_7825_wf__set,axiom,
    ! [A: $tType,Xs: list(product_prod(A,A))] :
      ( wf(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs))
    <=> transitive_acyclic(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) ) ).

% wf_set
tff(fact_7826_acyclicI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X4: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),transitive_trancl(A,R2)))
     => transitive_acyclic(A,R2) ) ).

% acyclicI
tff(fact_7827_acyclic__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( transitive_acyclic(A,R2)
    <=> ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),transitive_trancl(A,R2))) ) ).

% acyclic_def
tff(fact_7828_acyclicI__order,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [R2: set(product_prod(B,B)),F2: fun(B,A)] :
          ( ! [A4: B,B4: B] :
              ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B4)),R2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,B4)),aa(B,A,F2,A4))) )
         => transitive_acyclic(B,R2) ) ) ).

% acyclicI_order
tff(fact_7829_list_Osimps_I7_J,axiom,
    ! [C: $tType,A: $tType,F1: C,F22: fun(A,fun(list(A),fun(C,C))),X21: A,X222: list(A)] : aa(list(A),C,aa(fun(A,fun(list(A),fun(C,C))),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),rec_list(C,A),F1),F22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(C,C,aa(list(A),fun(C,C),aa(A,fun(list(A),fun(C,C)),F22,X21),X222),aa(list(A),C,aa(fun(A,fun(list(A),fun(C,C))),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),rec_list(C,A),F1),F22),X222)) ).

% list.simps(7)
tff(fact_7830_list_Osimps_I6_J,axiom,
    ! [A: $tType,C: $tType,F1: C,F22: fun(A,fun(list(A),fun(C,C)))] : aa(list(A),C,aa(fun(A,fun(list(A),fun(C,C))),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),rec_list(C,A),F1),F22),nil(A)) = F1 ).

% list.simps(6)
tff(fact_7831_list_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,fun(list(B),fun(C,C))),F2: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),C),comp(list(B),C,list(A),aa(fun(B,fun(list(B),fun(C,C))),fun(list(B),C),aa(C,fun(fun(B,fun(list(B),fun(C,C))),fun(list(B),C)),rec_list(C,B),G),Ga)),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2)) = aa(fun(A,fun(list(A),fun(C,C))),fun(list(A),C),aa(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),rec_list(C,A),G),aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_ake(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Ga),F2)) ).

% list.rec_o_map
tff(fact_7832_append__butlast__last__id,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),butlast(A),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),last(A,Xs)),nil(A))) = Xs ) ) ).

% append_butlast_last_id
tff(fact_7833_set__rec,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),aa(fun(A,fun(list(A),fun(set(A),set(A)))),fun(list(A),set(A)),aa(set(A),fun(fun(A,fun(list(A),fun(set(A),set(A)))),fun(list(A),set(A))),rec_list(set(A),A),bot_bot(set(A))),aTP_Lamp_akf(A,fun(list(A),fun(set(A),set(A))))),Xs) ).

% set_rec
tff(fact_7834_last__remdups__adj,axiom,
    ! [A: $tType,Xs: list(A)] : last(A,remdups_adj(A,Xs)) = last(A,Xs) ).

% last_remdups_adj
tff(fact_7835_last__appendR,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( ( Ys != nil(A) )
     => ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Ys) ) ) ).

% last_appendR
tff(fact_7836_last__appendL,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( ( Ys = nil(A) )
     => ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Xs) ) ) ).

% last_appendL
tff(fact_7837_last__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( last(A,aa(A,list(A),aa(nat,fun(A,list(A)),replicate(A),N),X)) = X ) ) ).

% last_replicate
tff(fact_7838_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( last(nat,upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_7839_last__snoc,axiom,
    ! [A: $tType,Xs: list(A),X: A] : last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = X ).

% last_snoc
tff(fact_7840_last__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( last(A,aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),N),Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_7841_last__rev,axiom,
    ! [A: $tType,Xs: list(A)] : last(A,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),A,hd(A),Xs) ).

% last_rev
tff(fact_7842_hd__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),A,hd(A),aa(list(A),list(A),rev(A),Xs)) = last(A,Xs) ).

% hd_rev
tff(fact_7843_last__in__set,axiom,
    ! [A: $tType,As: list(A)] :
      ( ( As != nil(A) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),last(A,As)),aa(list(A),set(A),set2(A),As))) ) ).

% last_in_set
tff(fact_7844_hd__Nil__eq__last,axiom,
    ! [A: $tType] : aa(list(A),A,hd(A),nil(A)) = last(A,nil(A)) ).

% hd_Nil_eq_last
tff(fact_7845_last_Osimps,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( ( Xs = nil(A) )
       => ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = X ) )
      & ( ( Xs != nil(A) )
       => ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = last(A,Xs) ) ) ) ).

% last.simps
tff(fact_7846_last__ConsL,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( Xs = nil(A) )
     => ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = X ) ) ).

% last_ConsL
tff(fact_7847_last__ConsR,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( Xs != nil(A) )
     => ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = last(A,Xs) ) ) ).

% last_ConsR
tff(fact_7848_last__append,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( ( ( Ys = nil(A) )
       => ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Xs) ) )
      & ( ( Ys != nil(A) )
       => ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Ys) ) ) ) ).

% last_append
tff(fact_7849_longest__common__suffix,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
    ? [Ss: list(A),Xs4: list(A),Ys5: list(A)] :
      ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),Ss) )
      & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),Ss) )
      & ( ( Xs4 = nil(A) )
        | ( Ys5 = nil(A) )
        | ( last(A,Xs4) != last(A,Ys5) ) ) ) ).

% longest_common_suffix
tff(fact_7850_last__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( ( Xs = nil(A) )
        | ( aa(list(A),list(A),tl(A),Xs) != nil(A) ) )
     => ( last(A,aa(list(A),list(A),tl(A),Xs)) = last(A,Xs) ) ) ).

% last_tl
tff(fact_7851_last__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B)] :
      ( ( Xs != nil(A) )
     => ( last(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs)) = aa(A,B,F2,last(A,Xs)) ) ) ).

% last_map
tff(fact_7852_dropWhile__last,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P,X))
       => ( last(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),P),Xs)) = last(A,Xs) ) ) ) ).

% dropWhile_last
tff(fact_7853_last__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
         => ( last(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys)) ) ) ) ) ).

% last_zip
tff(fact_7854_takeWhile__not__last,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),aTP_Lamp_akg(list(A),fun(A,bool),Xs)),Xs) = aa(list(A),list(A),butlast(A),Xs) ) ) ).

% takeWhile_not_last
tff(fact_7855_snoc__eq__iff__butlast,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = Ys )
    <=> ( ( Ys != nil(A) )
        & ( aa(list(A),list(A),butlast(A),Ys) = Xs )
        & ( last(A,Ys) = X ) ) ) ).

% snoc_eq_iff_butlast
tff(fact_7856_remdups__adj__append_H,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( ( Xs = nil(A) )
        | ( Ys = nil(A) )
        | ( last(A,Xs) != aa(list(A),A,hd(A),Ys) ) )
     => ( remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,Xs)),remdups_adj(A,Ys)) ) ) ).

% remdups_adj_append'
tff(fact_7857_remdups__adj__append_H_H,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,Xs)),remdups_adj(A,aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),dropWhile(A),aTP_Lamp_akh(list(A),fun(A,bool),Xs)),Ys))) ) ) ).

% remdups_adj_append''
tff(fact_7858_last__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( last(A,Xs) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))) ) ) ).

% last_conv_nth
tff(fact_7859_last__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,X: A] :
      ( ( Xs != nil(A) )
     => ( ( ( K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
         => ( last(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X)) = X ) )
        & ( ( K != aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
         => ( last(A,aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Xs),K),X)) = last(A,Xs) ) ) ) ) ).

% last_list_update
tff(fact_7860_List_Ounion__def,axiom,
    ! [A: $tType] : union(A) = aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(list(A),list(A))),fold(A,list(A)),insert(A)) ).

% List.union_def
tff(fact_7861_connected__closed,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( topolo1966860045006549960nected(A,S)
        <=> ~ ? [A8: set(A),B10: set(A)] :
                ( topolo7761053866217962861closed(A,A8)
                & topolo7761053866217962861closed(A,B10)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A8),B10)))
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),B10)),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)),B10),S) != bot_bot(set(A)) ) ) ) ) ).

% connected_closed
tff(fact_7862_in__set__insert,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs) = Xs ) ) ).

% in_set_insert
tff(fact_7863_distinct__insert,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs))
    <=> distinct(A,Xs) ) ).

% distinct_insert
tff(fact_7864_insert__Nil,axiom,
    ! [A: $tType,X: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).

% insert_Nil
tff(fact_7865_not__in__set__insert,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) ) ).

% not_in_set_insert
tff(fact_7866_List_Oset__insert,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs)) ).

% List.set_insert
tff(fact_7867_connected__Times__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S2: set(A),T3: set(B)] :
          ( topolo1966860045006549960nected(product_prod(A,B),product_Sigma(A,B,S2,aTP_Lamp_aki(set(B),fun(A,set(B)),T3)))
        <=> ( ( S2 = bot_bot(set(A)) )
            | ( T3 = bot_bot(set(B)) )
            | ( topolo1966860045006549960nected(A,S2)
              & topolo1966860045006549960nected(B,T3) ) ) ) ) ).

% connected_Times_eq
tff(fact_7868_connected__contains__Ioo,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: set(A),A2: A,B2: A] :
          ( topolo1966860045006549960nected(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),A3)) ) ) ) ) ).

% connected_contains_Ioo
tff(fact_7869_connected__iff__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U2: set(A)] :
          ( topolo1966860045006549960nected(A,U2)
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),U2))
             => ! [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),U2))
                 => ! [Z2: A] :
                      ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2))
                     => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Xa3))
                       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),U2)) ) ) ) ) ) ) ).

% connected_iff_interval
tff(fact_7870_connectedI__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U2: set(A)] :
          ( ! [X4: A,Y2: A,Z3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),U2))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),U2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Z3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),Y2))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),U2)) ) ) ) )
         => topolo1966860045006549960nected(A,U2) ) ) ).

% connectedI_interval
tff(fact_7871_connectedD__interval,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [U2: set(A),X: A,Y: A,Z: A] :
          ( topolo1966860045006549960nected(A,U2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),U2))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),U2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),U2)) ) ) ) ) ) ) ).

% connectedD_interval
tff(fact_7872_connected__contains__Icc,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: set(A),A2: A,B2: A] :
          ( topolo1966860045006549960nected(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),A3)) ) ) ) ) ).

% connected_contains_Icc
tff(fact_7873_connected__sing,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A] : topolo1966860045006549960nected(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) ) ).

% connected_sing
tff(fact_7874_connected__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1966860045006549960nected(A,bot_bot(set(A))) ) ).

% connected_empty
tff(fact_7875_insert__remdups,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),remdups(A,Xs)) = remdups(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs)) ).

% insert_remdups
tff(fact_7876_connected__Un,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T2: set(A)] :
          ( topolo1966860045006549960nected(A,S)
         => ( topolo1966860045006549960nected(A,T2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2) != bot_bot(set(A)) )
             => topolo1966860045006549960nected(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T2)) ) ) ) ) ).

% connected_Un
tff(fact_7877_List_Oinsert__def,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs) = Xs ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) ) ) ).

% List.insert_def
tff(fact_7878_connected__Union,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(set(A))] :
          ( ! [S4: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),S4),S2))
             => topolo1966860045006549960nected(A,S4) )
         => ( ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S2) != bot_bot(set(A)) )
           => topolo1966860045006549960nected(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S2)) ) ) ) ).

% connected_Union
tff(fact_7879_not__in__connected__cases,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),X: A] :
          ( topolo1966860045006549960nected(A,S2)
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
           => ( ( S2 != bot_bot(set(A)) )
             => ( ( condit941137186595557371_above(A,S2)
                 => ~ ! [Y3: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),S2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) ) )
               => ~ ( condit1013018076250108175_below(A,S2)
                   => ~ ! [Y3: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),S2))
                         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) ) ) ) ) ) ) ) ).

% not_in_connected_cases
tff(fact_7880_connected__diff__open__from__closed,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T2: set(A),U: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),T2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),U))
           => ( topolo1002775350975398744n_open(A,S)
             => ( topolo7761053866217962861closed(A,T2)
               => ( topolo1966860045006549960nected(A,U)
                 => ( topolo1966860045006549960nected(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                   => topolo1966860045006549960nected(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),U),S)) ) ) ) ) ) ) ) ).

% connected_diff_open_from_closed
tff(fact_7881_remove__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),coset(A,Xs)) = coset(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),X),Xs)) ).

% remove_code(2)
tff(fact_7882_connected__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo1966860045006549960nected(A,S2)
        <=> ~ ? [A8: set(A),B10: set(A)] :
                ( topolo1002775350975398744n_open(A,A8)
                & topolo1002775350975398744n_open(A,B10)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A8),B10)))
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),B10)),S2) = bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),S2) != bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B10),S2) != bot_bot(set(A)) ) ) ) ) ).

% connected_def
tff(fact_7883_connectedI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U2: set(A)] :
          ( ! [A5: set(A)] :
              ( topolo1002775350975398744n_open(A,A5)
             => ! [B8: set(A)] :
                  ( topolo1002775350975398744n_open(A,B8)
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),U2) != bot_bot(set(A)) )
                   => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B8),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)),A5),B8)),U2) = bot_bot(set(A)) )
                       => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),U2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B8))) ) ) ) ) )
         => topolo1966860045006549960nected(A,U2) ) ) ).

% connectedI
tff(fact_7884_connectedD,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: set(A),U2: set(A),V: set(A)] :
          ( topolo1966860045006549960nected(A,A3)
         => ( 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)),A3) = bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),U2),V)))
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),A3) = bot_bot(set(A)) )
                    | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),V),A3) = bot_bot(set(A)) ) ) ) ) ) ) ) ) ).

% connectedD
tff(fact_7885_connected__closedD,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),A3: set(A),B3: set(A)] :
          ( topolo1966860045006549960nected(A,S)
         => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),S) = bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))
             => ( topolo7761053866217962861closed(A,A3)
               => ( topolo7761053866217962861closed(A,B3)
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),S) = bot_bot(set(A)) )
                    | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),S) = bot_bot(set(A)) ) ) ) ) ) ) ) ) ).

% connected_closedD
tff(fact_7886_irreflp__irrefl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irreflp(A,aTP_Lamp_wl(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
    <=> irrefl(A,R) ) ).

% irreflp_irrefl_eq
tff(fact_7887_disjnt__equiv__class,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))))
      <=> ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) ) ) ).

% disjnt_equiv_class
tff(fact_7888_disjnt__self__iff__empty,axiom,
    ! [A: $tType,S2: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),S2),S2))
    <=> ( S2 = bot_bot(set(A)) ) ) ).

% disjnt_self_iff_empty
tff(fact_7889_disjnt__insert1,axiom,
    ! [A: $tType,A2: A,X6: set(A),Y5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X6)),Y5))
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),Y5))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y5)) ) ) ).

% disjnt_insert1
tff(fact_7890_disjnt__insert2,axiom,
    ! [A: $tType,Y5: set(A),A2: A,X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Y5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X6)))
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),Y5))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Y5),X6)) ) ) ).

% disjnt_insert2
tff(fact_7891_disjnt__Un2,axiom,
    ! [A: $tType,C3: set(A),A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C3),A3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C3),B3)) ) ) ).

% disjnt_Un2
tff(fact_7892_disjnt__Un1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A3),C3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),B3),C3)) ) ) ).

% disjnt_Un1
tff(fact_7893_disjnt__Times2__iff,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C3: set(B),B3: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_agg(set(B),fun(A,set(B)),C3))),product_Sigma(A,B,B3,aTP_Lamp_agg(set(B),fun(A,set(B)),C3))))
    <=> ( ( C3 = bot_bot(set(B)) )
        | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A3),B3)) ) ) ).

% disjnt_Times2_iff
tff(fact_7894_disjnt__Times1__iff,axiom,
    ! [A: $tType,B: $tType,C3: set(A),A3: set(B),B3: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,C3,aTP_Lamp_agg(set(B),fun(A,set(B)),A3))),product_Sigma(A,B,C3,aTP_Lamp_agg(set(B),fun(A,set(B)),B3))))
    <=> ( ( C3 = bot_bot(set(A)) )
        | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),A3),B3)) ) ) ).

% disjnt_Times1_iff
tff(fact_7895_irreflp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,aTP_Lamp_akj(A,fun(A,bool))) ) ).

% irreflp_greater
tff(fact_7896_disjnt__sym,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A3),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),B3),A3)) ) ).

% disjnt_sym
tff(fact_7897_disjnt__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A3),B3))
    <=> ! [X3: A] :
          ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
            & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3)) ) ) ).

% disjnt_iff
tff(fact_7898_irreflp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,ord_less(A)) ) ).

% irreflp_less
tff(fact_7899_disjnt__empty2,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A3),bot_bot(set(A)))) ).

% disjnt_empty2
tff(fact_7900_disjnt__empty1,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),bot_bot(set(A))),A3)) ).

% disjnt_empty1
tff(fact_7901_disjnt__insert,axiom,
    ! [A: $tType,X: A,N4: set(A),M4: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),N4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),M4),N4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),M4)),N4)) ) ) ).

% disjnt_insert
tff(fact_7902_disjnt__subset2,axiom,
    ! [A: $tType,X6: set(A),Y5: set(A),Z5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z5),Y5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Z5)) ) ) ).

% disjnt_subset2
tff(fact_7903_disjnt__subset1,axiom,
    ! [A: $tType,X6: set(A),Y5: set(A),Z5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z5),X6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Z5),Y5)) ) ) ).

% disjnt_subset1
tff(fact_7904_disjnt__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C3: fun(A,set(B)),B3: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A3,C3)),product_Sigma(A,B,B3,C3)))
    <=> ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))
           => ( aa(A,set(B),C3,X3) = bot_bot(set(B)) ) )
        | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A3),B3)) ) ) ).

% disjnt_Sigma_iff
tff(fact_7905_disjnt__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A3),B3))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ).

% disjnt_def
tff(fact_7906_disjnt__ge__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y5: set(A),X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),Y5))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),Y5)),X4)) )
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y5)) ) ) ) ).

% disjnt_ge_max
tff(fact_7907_card__Un__disjnt,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pp(aa(set(A),bool,finite_finite2(A),B3))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A3),B3))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ) ).

% card_Un_disjnt
tff(fact_7908_sum__card__image,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A3))
     => ( pairwise(A,aTP_Lamp_akk(fun(A,set(B)),fun(A,fun(A,bool)),F2),A3)
       => ( 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),A3)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ma(fun(A,set(B)),fun(A,nat),F2)),A3) ) ) ) ).

% sum_card_image
tff(fact_7909_infinite__infinite__partition,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A3))
     => ~ ! [C7: fun(nat,set(A))] :
            ( pairwise(nat,aTP_Lamp_akl(fun(nat,set(A)),fun(nat,fun(nat,bool)),C7),top_top(set(nat)))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),C7),top_top(set(nat))))),A3))
             => ~ ! [I3: nat] : ~ pp(aa(set(A),bool,finite_finite2(A),aa(nat,set(A),C7,I3))) ) ) ) ).

% infinite_infinite_partition
tff(fact_7910_pairwise__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),P: fun(B,fun(B,bool))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),A3))
           => ( ( X4 != Y2 )
             => ( ( aa(A,B,F2,X4) != aa(A,B,F2,Y2) )
               => pp(aa(B,bool,aa(B,fun(B,bool),P,aa(A,B,F2,X4)),aa(A,B,F2,Y2))) ) ) ) )
     => pairwise(B,P,aa(set(A),set(B),image2(A,B,F2),A3)) ) ).

% pairwise_imageI
tff(fact_7911_pairwise__image,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,bool)),F2: fun(B,A),S: set(B)] :
      ( pairwise(A,R2,aa(set(B),set(A),image2(B,A,F2),S))
    <=> pairwise(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_akm(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R2),F2),S) ) ).

% pairwise_image
tff(fact_7912_pairwise__mono,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),A3: set(A),Q: fun(A,fun(A,bool)),B3: set(A)] :
      ( pairwise(A,P,A3)
     => ( ! [X4: A,Y2: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Y2))
           => pp(aa(A,bool,aa(A,fun(A,bool),Q,X4),Y2)) )
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
         => pairwise(A,Q,B3) ) ) ) ).

% pairwise_mono
tff(fact_7913_pairwise__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),S2: set(A),T3: set(A)] :
      ( pairwise(A,P,S2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),S2))
       => pairwise(A,P,T3) ) ) ).

% pairwise_subset
tff(fact_7914_pairwise__insert,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X: A,S: set(A)] :
      ( pairwise(A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),S))
    <=> ( ! [Y4: A] :
            ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S))
              & ( Y4 != X ) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X),Y4))
              & pp(aa(A,bool,aa(A,fun(A,bool),R2,Y4),X)) ) )
        & pairwise(A,R2,S) ) ) ).

% pairwise_insert
tff(fact_7915_pairwise__singleton,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),A3: A] : pairwise(A,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ).

% pairwise_singleton
tff(fact_7916_pairwise__empty,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : pairwise(A,P,bot_bot(set(A))) ).

% pairwise_empty
tff(fact_7917_pairwiseD,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),S2: set(A),X: A,Y: A] :
      ( pairwise(A,R,S2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
         => ( ( X != Y )
           => pp(aa(A,bool,aa(A,fun(A,bool),R,X),Y)) ) ) ) ) ).

% pairwiseD
tff(fact_7918_pairwiseI,axiom,
    ! [A: $tType,S2: set(A),R: fun(A,fun(A,bool))] :
      ( ! [X4: A,Y2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y2),S2))
           => ( ( X4 != Y2 )
             => pp(aa(A,bool,aa(A,fun(A,bool),R,X4),Y2)) ) ) )
     => pairwise(A,R,S2) ) ).

% pairwiseI
tff(fact_7919_pairwise__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),S2: set(A)] :
      ( pairwise(A,R,S2)
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),S2))
             => ( ( X3 != Xa3 )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,X3),Xa3)) ) ) ) ) ).

% pairwise_def
tff(fact_7920_pairwise__trivial,axiom,
    ! [A: $tType,I6: set(A)] : pairwise(A,aTP_Lamp_aeo(A,fun(A,bool)),I6) ).

% pairwise_trivial
tff(fact_7921_pairwise__alt,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),S2: set(A)] :
      ( pairwise(A,R,S2)
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))))
             => pp(aa(A,bool,aa(A,fun(A,bool),R,X3),Xa3)) ) ) ) ).

% pairwise_alt
tff(fact_7922_disjoint__image__subset,axiom,
    ! [A: $tType,A19: set(set(A)),F2: fun(set(A),set(A))] :
      ( pairwise(set(A),disjnt(A),A19)
     => ( ! [X7: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X7),A19))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),F2,X7)),X7)) )
       => pairwise(set(A),disjnt(A),aa(set(set(A)),set(set(A)),image2(set(A),set(A),F2),A19)) ) ) ).

% disjoint_image_subset
tff(fact_7923_card__Union__disjoint,axiom,
    ! [A: $tType,C3: set(set(A))] :
      ( pairwise(set(A),disjnt(A),C3)
     => ( ! [A5: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A5),C3))
           => pp(aa(set(A),bool,finite_finite2(A),A5)) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C3)) = aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),C3) ) ) ) ).

% card_Union_disjoint
tff(fact_7924_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))),bool),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)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),bool),aTP_Lamp_akn(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)))) ).

% init_seg_of_def
tff(fact_7925_map__tailrec__rev,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),As: list(B),Bs: list(A)] : map_tailrec_rev(B,A,F2,As,Bs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),As))),Bs) ).

% map_tailrec_rev
tff(fact_7926_map__tailrec__rev_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),Bs: list(B)] : map_tailrec_rev(A,B,F2,nil(A),Bs) = Bs ).

% map_tailrec_rev.simps(1)
tff(fact_7927_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A2: A,As: list(A),Bs: list(B)] : map_tailrec_rev(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),As),Bs) = map_tailrec_rev(A,B,F2,As,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,F2,A2)),Bs)) ).

% map_tailrec_rev.simps(2)
tff(fact_7928_trans__init__seg__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A)),T2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),S)),init_seg_of(A)))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),S),T2)),init_seg_of(A)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),T2)),init_seg_of(A))) ) ) ).

% trans_init_seg_of
tff(fact_7929_antisym__init__seg__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),S)),init_seg_of(A)))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),S),R2)),init_seg_of(A)))
       => ( R2 = S ) ) ) ).

% antisym_init_seg_of
tff(fact_7930_refl__on__init__seg__of,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),R2)),init_seg_of(A))) ).

% refl_on_init_seg_of
tff(fact_7931_map__tailrec__rev_Oelims,axiom,
    ! [A: $tType,B: $tType,X: fun(A,B),Xa2: list(A),Xb2: list(B),Y: list(B)] :
      ( ( map_tailrec_rev(A,B,X,Xa2,Xb2) = Y )
     => ( ( ( Xa2 = nil(A) )
         => ( Y != Xb2 ) )
       => ~ ! [A4: A,As2: list(A)] :
              ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As2) )
             => ( Y != map_tailrec_rev(A,B,X,As2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,X,A4)),Xb2)) ) ) ) ) ).

% map_tailrec_rev.elims
tff(fact_7932_Chains__init__seg__of__Union,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A))),R2: set(product_prod(A,A))] :
      ( pp(aa(set(set(set(product_prod(A,A)))),bool,aa(set(set(product_prod(A,A))),fun(set(set(set(product_prod(A,A)))),bool),member(set(set(product_prod(A,A)))),R),chains(set(product_prod(A,A)),init_seg_of(A))))
     => ( pp(aa(set(set(product_prod(A,A))),bool,aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),bool),member(set(product_prod(A,A))),R2),R))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R))),init_seg_of(A))) ) ) ).

% Chains_init_seg_of_Union
tff(fact_7933_initial__segment__of__Diff,axiom,
    ! [A: $tType,P2: set(product_prod(A,A)),Q4: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),P2),Q4)),init_seg_of(A)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),P2),S)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Q4),S))),init_seg_of(A))) ) ).

% initial_segment_of_Diff
tff(fact_7934_map__tailrec__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),As: list(A)] : aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map_tailrec(A,B),F2),As) = aa(list(B),list(B),rev(B),map_tailrec_rev(A,B,F2,As,nil(B))) ).

% map_tailrec_def
tff(fact_7935_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),N))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) )
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_not_iff_eq
tff(fact_7936_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% not_nonnegative_int_iff
tff(fact_7937_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% not_negative_int_iff
tff(fact_7938_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_7939_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_7940_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_7941_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_7942_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_7943_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q4: num] :
      ( ( bit_and_not_num(M,N) = aa(num,option(num),some(num),Q4) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(num,int,numeral_numeral(int),Q4) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_7944_map__eq__map__tailrec,axiom,
    ! [B: $tType,A: $tType] : map(A,B) = map_tailrec(A,B) ).

% map_eq_map_tailrec
tff(fact_7945_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_7946_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_7947_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_7948_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_7949_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_7950_insort__insert__key__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X)),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))))
           => ( linord329482645794927042rt_key(B,A,F2,X,Xs) = Xs ) )
          & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X)),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))))
           => ( linord329482645794927042rt_key(B,A,F2,X,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs) ) ) ) ) ).

% insort_insert_key_def
tff(fact_7951_insort__insert__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] :
          ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X)),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))))
         => ( linord329482645794927042rt_key(B,A,F2,X,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),X),Xs) ) ) ) ).

% insort_insert_insort_key
tff(fact_7952_distinct__insort__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Xs: list(B),F2: fun(B,A),X: B] :
          ( distinct(B,Xs)
         => distinct(B,linord329482645794927042rt_key(B,A,F2,X,Xs)) ) ) ).

% distinct_insort_insert
tff(fact_7953_insort__insert__triv,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_kg(A,A),X,Xs) = Xs ) ) ) ).

% insort_insert_triv
tff(fact_7954_insort__insert__key__triv,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),X: B,Xs: list(B)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F2,X)),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))))
         => ( linord329482645794927042rt_key(B,A,F2,X,Xs) = Xs ) ) ) ).

% insort_insert_key_triv
tff(fact_7955_set__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),linord329482645794927042rt_key(A,A,aTP_Lamp_kg(A,A),X,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_insert
tff(fact_7956_sorted__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),X: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_kg(A,A),X,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_7957_insort__insert__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_kg(A,A),X,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_kg(A,A)),X),Xs) ) ) ) ).

% insort_insert_insort
tff(fact_7958_sorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),linord329482645794927042rt_key(B,A,F2,X,Xs))) ) ) ).

% sorted_insort_insert_key
tff(fact_7959_Real_Opositive_Oabs__eq,axiom,
    ! [X: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X),X))
     => ( pp(aa(real,bool,positive2,real2(X)))
      <=> ? [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
            & ? [K3: nat] :
              ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N3))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,X,N3))) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_7960_arg__min__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F2: fun(A,B),P: fun(A,bool),A2: A] :
          ( inj_on(A,B,F2,aa(fun(A,bool),set(A),collect(A),P))
         => ( pp(aa(A,bool,P,A2))
           => ( ! [Y2: A] :
                  ( pp(aa(A,bool,P,Y2))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,A2)),aa(A,B,F2,Y2))) )
             => ( lattices_ord_arg_min(A,B,F2,P) = A2 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_7961_arg__minI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [P: fun(A,bool),X: A,F2: fun(A,B),Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y2: A] :
                ( pp(aa(A,bool,P,Y2))
               => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X))) )
           => ( ! [X4: A] :
                  ( pp(aa(A,bool,P,X4))
                 => ( ! [Y3: A] :
                        ( pp(aa(A,bool,P,Y3))
                       => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F2,Y3)),aa(A,B,F2,X4))) )
                   => pp(aa(A,bool,Q,X4)) ) )
             => pp(aa(A,bool,Q,lattices_ord_arg_min(A,B,F2,P))) ) ) ) ) ).

% arg_minI
tff(fact_7962_arg__min__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => ( pp(aa(A,bool,P,lattices_ord_arg_min(A,nat,M,P)))
        & ! [Y3: A] :
            ( pp(aa(A,bool,P,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,Y3))) ) ) ) ).

% arg_min_nat_lemma
tff(fact_7963_arg__min__nat__le,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,X))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,X))) ) ).

% arg_min_nat_le
tff(fact_7964_arg__min__equality,axiom,
    ! [A: $tType,C: $tType] :
      ( order(A)
     => ! [P: fun(C,bool),K: C,F2: fun(C,A)] :
          ( pp(aa(C,bool,P,K))
         => ( ! [X4: C] :
                ( pp(aa(C,bool,P,X4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F2,K)),aa(C,A,F2,X4))) )
           => ( aa(C,A,F2,lattices_ord_arg_min(C,A,F2,P)) = aa(C,A,F2,K) ) ) ) ) ).

% arg_min_equality
tff(fact_7965_arg__min__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F2: fun(B,A),S2: set(B)] : lattic7623131987881927897min_on(B,A,F2,S2) = lattices_ord_arg_min(B,A,F2,aTP_Lamp_tf(set(B),fun(B,bool),S2)) ) ).

% arg_min_on_def
tff(fact_7966_arg__min__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F2: fun(B,A),P: fun(B,bool)] : lattices_ord_arg_min(B,A,F2,P) = fChoice(B,lattic501386751177426532rg_min(B,A,F2,P)) ) ).

% arg_min_def
tff(fact_7967_Real_Opositive_Orsp,axiom,
    pp(aa(fun(fun(nat,rat),bool),bool,aa(fun(fun(nat,rat),bool),fun(fun(fun(nat,rat),bool),bool),bNF_rel_fun(fun(nat,rat),fun(nat,rat),bool,bool,realrel,fequal(bool)),aTP_Lamp_ako(fun(nat,rat),bool)),aTP_Lamp_ako(fun(nat,rat),bool))) ).

% Real.positive.rsp
tff(fact_7968_arg__min__natI,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => pp(aa(A,bool,P,lattices_ord_arg_min(A,nat,M,P))) ) ).

% arg_min_natI
tff(fact_7969_is__arg__min__arg__min__nat,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,X))
     => pp(aa(A,bool,lattic501386751177426532rg_min(A,nat,M,P),lattices_ord_arg_min(A,nat,M,P))) ) ).

% is_arg_min_arg_min_nat
tff(fact_7970_list_Orec__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,S2: fun(C,fun(D,bool)),R: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,fun(list(B),fun(D,D))),fun(list(B),D))),bool,aa(fun(C,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C))),fun(fun(D,fun(fun(B,fun(list(B),fun(D,D))),fun(list(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,fun(list(A),fun(C,C))),fun(list(A),C)),fun(fun(B,fun(list(B),fun(D,D))),fun(list(B),D)),S2,bNF_rel_fun(fun(A,fun(list(A),fun(C,C))),fun(B,fun(list(B),fun(D,D))),fun(list(A),C),fun(list(B),D),bNF_rel_fun(A,B,fun(list(A),fun(C,C)),fun(list(B),fun(D,D)),R,bNF_rel_fun(list(A),list(B),fun(C,C),fun(D,D),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),bNF_rel_fun(C,D,C,D,S2,S2))),bNF_rel_fun(list(A),list(B),C,D,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),S2))),rec_list(C,A)),rec_list(D,B))) ).

% list.rec_transfer
tff(fact_7971_foldr__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,A3: fun(A,fun(C,bool)),B3: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,D)),fun(list(C),fun(D,D))),bool,aa(fun(fun(A,fun(B,B)),fun(list(A),fun(B,B))),fun(fun(fun(C,fun(D,D)),fun(list(C),fun(D,D))),bool),bNF_rel_fun(fun(A,fun(B,B)),fun(C,fun(D,D)),fun(list(A),fun(B,B)),fun(list(C),fun(D,D)),bNF_rel_fun(A,C,fun(B,B),fun(D,D),A3,bNF_rel_fun(B,D,B,D,B3,B3)),bNF_rel_fun(list(A),list(C),fun(B,B),fun(D,D),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),A3),bNF_rel_fun(B,D,B,D,B3,B3))),foldr(A,B)),foldr(C,D))) ).

% foldr_transfer
tff(fact_7972_List_Ofold__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,A3: fun(A,fun(C,bool)),B3: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,D)),fun(list(C),fun(D,D))),bool,aa(fun(fun(A,fun(B,B)),fun(list(A),fun(B,B))),fun(fun(fun(C,fun(D,D)),fun(list(C),fun(D,D))),bool),bNF_rel_fun(fun(A,fun(B,B)),fun(C,fun(D,D)),fun(list(A),fun(B,B)),fun(list(C),fun(D,D)),bNF_rel_fun(A,C,fun(B,B),fun(D,D),A3,bNF_rel_fun(B,D,B,D,B3,B3)),bNF_rel_fun(list(A),list(C),fun(B,B),fun(D,D),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),A3),bNF_rel_fun(B,D,B,D,B3,B3))),fold(A,B)),fold(C,D))) ).

% List.fold_transfer
tff(fact_7973_horner__sum__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [A3: fun(A,fun(B,bool)),B3: fun(C,fun(D,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A3,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A3,bNF_rel_fun(A,B,A,B,A3,A3)),plus_plus(A)),plus_plus(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A3,bNF_rel_fun(A,B,A,B,A3,A3)),times_times(A)),times_times(B)))
             => pp(aa(fun(fun(D,B),fun(B,fun(list(D),B))),bool,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),bool),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B3,A3),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A3,bNF_rel_fun(list(C),list(D),A,B,aa(fun(C,fun(D,bool)),fun(list(C),fun(list(D),bool)),list_all2(C,D),B3),A3))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B))) ) ) ) ) ).

% horner_sum_transfer
tff(fact_7974_list_Omap__transfer,axiom,
    ! [A: $tType,B: $tType,F: $tType,E: $tType,Rb2: fun(A,fun(E,bool)),Sd: fun(B,fun(F,bool))] : pp(aa(fun(fun(E,F),fun(list(E),list(F))),bool,aa(fun(fun(A,B),fun(list(A),list(B))),fun(fun(fun(E,F),fun(list(E),list(F))),bool),bNF_rel_fun(fun(A,B),fun(E,F),fun(list(A),list(B)),fun(list(E),list(F)),bNF_rel_fun(A,E,B,F,Rb2,Sd),bNF_rel_fun(list(A),list(E),list(B),list(F),aa(fun(A,fun(E,bool)),fun(list(A),fun(list(E),bool)),list_all2(A,E),Rb2),aa(fun(B,fun(F,bool)),fun(list(B),fun(list(F),bool)),list_all2(B,F),Sd))),map(A,B)),map(E,F))) ).

% list.map_transfer
tff(fact_7975_foldl__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,B3: fun(A,fun(C,bool)),A3: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,C)),fun(C,fun(list(D),C))),bool,aa(fun(fun(A,fun(B,A)),fun(A,fun(list(B),A))),fun(fun(fun(C,fun(D,C)),fun(C,fun(list(D),C))),bool),bNF_rel_fun(fun(A,fun(B,A)),fun(C,fun(D,C)),fun(A,fun(list(B),A)),fun(C,fun(list(D),C)),bNF_rel_fun(A,C,fun(B,A),fun(D,C),B3,bNF_rel_fun(B,D,A,C,A3,B3)),bNF_rel_fun(A,C,fun(list(B),A),fun(list(D),C),B3,bNF_rel_fun(list(B),list(D),A,C,aa(fun(B,fun(D,bool)),fun(list(B),fun(list(D),bool)),list_all2(B,D),A3),B3))),foldl(A,B)),foldl(C,D))) ).

% foldl_transfer
tff(fact_7976_sum__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & monoid_add(A) )
     => ! [A3: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A3,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A3,bNF_rel_fun(A,B,A,B,A3,A3)),plus_plus(A)),plus_plus(B)))
           => pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),bNF_rel_fun(list(A),list(B),A,B,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),A3),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B))) ) ) ) ).

% sum_list_transfer
tff(fact_7977_is__arg__min__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F2: fun(A,B),P: fun(A,bool),X: A,Y: A] :
          ( pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F2,P),X))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,X)))
           => ( pp(aa(A,bool,P,Y))
             => pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F2,P),Y)) ) ) ) ) ).

% is_arg_min_antimono
tff(fact_7978_is__arg__min__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F2,P),X))
        <=> ( pp(aa(A,bool,P,X))
            & ! [Y4: A] :
                ( pp(aa(A,bool,P,Y4))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y4))) ) ) ) ) ).

% is_arg_min_linorder
tff(fact_7979_is__arg__min__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F2: fun(B,A),P: fun(B,bool),X: B] :
          ( pp(aa(B,bool,lattic501386751177426532rg_min(B,A,F2,P),X))
        <=> ( pp(aa(B,bool,P,X))
            & ~ ? [Y4: B] :
                  ( pp(aa(B,bool,P,Y4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F2,Y4)),aa(B,A,F2,X))) ) ) ) ) ).

% is_arg_min_def
tff(fact_7980_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B)))
             => ( pp(aa(fun(B,B),bool,aa(fun(A,A),fun(fun(B,B),bool),bNF_rel_fun(A,B,A,B,R,R),uminus_uminus(A)),uminus_uminus(B)))
               => pp(aa(fun(int,B),bool,aa(fun(int,A),fun(fun(int,B),bool),bNF_rel_fun(int,int,A,B,fequal(int),R),ring_1_of_int(A)),ring_1_of_int(B))) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_7981_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B)))
             => pp(aa(fun(num,B),bool,aa(fun(num,A),fun(fun(num,B),bool),bNF_rel_fun(num,num,A,B,fequal(num),R),numeral_numeral(A)),numeral_numeral(B))) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_7982_ex__is__arg__min__if__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F2: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X_1: A] : pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F2,aTP_Lamp_a(set(A),fun(A,bool),S2)),X_1)) ) ) ) ).

% ex_is_arg_min_if_finite
tff(fact_7983_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B)))
             => pp(aa(fun(nat,B),bool,aa(fun(nat,A),fun(fun(nat,B),bool),bNF_rel_fun(nat,nat,A,B,fequal(nat),R),semiring_1_of_nat(A)),semiring_1_of_nat(B))) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_7984_maps__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,list(B)),Xs: list(A)] : maps(A,B,F2,Xs) = aa(list(list(B)),list(B),concat(B),aa(list(A),list(list(B)),aa(fun(A,list(B)),fun(list(A),list(list(B))),map(A,list(B)),F2),Xs)) ).

% maps_def
tff(fact_7985_splice__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),fun(list(B),list(B))),bool,aa(fun(list(A),fun(list(A),list(A))),fun(fun(list(B),fun(list(B),list(B))),bool),bNF_rel_fun(list(A),list(B),fun(list(A),list(A)),fun(list(B),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),splice(A)),splice(B))) ).

% splice_transfer
tff(fact_7986_butlast__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3)),butlast(A)),butlast(B))) ).

% butlast_transfer
tff(fact_7987_dropWhile__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),list(B))),bool,aa(fun(fun(A,bool),fun(list(A),list(A))),fun(fun(fun(B,bool),fun(list(B),list(B))),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),list(A)),fun(list(B),list(B)),bNF_rel_fun(A,B,bool,bool,A3,fequal(bool)),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),dropWhile(A)),dropWhile(B))) ).

% dropWhile_transfer
tff(fact_7988_takeWhile__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),list(B))),bool,aa(fun(fun(A,bool),fun(list(A),list(A))),fun(fun(fun(B,bool),fun(list(B),list(B))),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),list(A)),fun(list(B),list(B)),bNF_rel_fun(A,B,bool,bool,A3,fequal(bool)),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),takeWhile(A)),takeWhile(B))) ).

% takeWhile_transfer
tff(fact_7989_List_Ofilter__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(list(B),list(B))),bool,aa(fun(fun(A,bool),fun(list(A),list(A))),fun(fun(fun(B,bool),fun(list(B),list(B))),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(list(A),list(A)),fun(list(B),list(B)),bNF_rel_fun(A,B,bool,bool,A3,fequal(bool)),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),filter2(A)),filter2(B))) ).

% List.filter_transfer
tff(fact_7990_tl__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3)),tl(A)),tl(B))) ).

% tl_transfer
tff(fact_7991_append__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),fun(list(B),list(B))),bool,aa(fun(list(A),fun(list(A),list(A))),fun(fun(list(B),fun(list(B),list(B))),bool),bNF_rel_fun(list(A),list(B),fun(list(A),list(A)),fun(list(B),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),append(A)),append(B))) ).

% append_transfer
tff(fact_7992_list_Octr__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(fun(B,fun(list(B),list(B))),bool,aa(fun(A,fun(list(A),list(A))),fun(fun(B,fun(list(B),list(B))),bool),bNF_rel_fun(A,B,fun(list(A),list(A)),fun(list(B),list(B)),R,bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R))),cons(A)),cons(B))) ).

% list.ctr_transfer(2)
tff(fact_7993_concat__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(list(B)),list(B)),bool,aa(fun(list(list(A)),list(A)),fun(fun(list(list(B)),list(B)),bool),bNF_rel_fun(list(list(A)),list(list(B)),list(A),list(B),aa(fun(list(A),fun(list(B),bool)),fun(list(list(A)),fun(list(list(B)),bool)),list_all2(list(A),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3)),concat(A)),concat(B))) ).

% concat_transfer
tff(fact_7994_replicate__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(nat,fun(B,list(B))),bool,aa(fun(nat,fun(A,list(A))),fun(fun(nat,fun(B,list(B))),bool),bNF_rel_fun(nat,nat,fun(A,list(A)),fun(B,list(B)),fequal(nat),bNF_rel_fun(A,B,list(A),list(B),A3,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),replicate(A)),replicate(B))) ).

% replicate_transfer
tff(fact_7995_product__lists__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(list(B)),list(list(B))),bool,aa(fun(list(list(A)),list(list(A))),fun(fun(list(list(B)),list(list(B))),bool),bNF_rel_fun(list(list(A)),list(list(B)),list(list(A)),list(list(B)),aa(fun(list(A),fun(list(B),bool)),fun(list(list(A)),fun(list(list(B)),bool)),list_all2(list(A),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3)),aa(fun(list(A),fun(list(B),bool)),fun(list(list(A)),fun(list(list(B)),bool)),list_all2(list(A),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),product_lists(A)),product_lists(B))) ).

% product_lists_transfer
tff(fact_7996_list_Orel__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,bool)),Sc: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,bool)),fun(list(C),fun(list(D),bool))),bool,aa(fun(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool))),fun(fun(fun(C,fun(D,bool)),fun(list(C),fun(list(D),bool))),bool),bNF_rel_fun(fun(A,fun(B,bool)),fun(C,fun(D,bool)),fun(list(A),fun(list(B),bool)),fun(list(C),fun(list(D),bool)),bNF_rel_fun(A,C,fun(B,bool),fun(D,bool),Sa,bNF_rel_fun(B,D,bool,bool,Sc,fequal(bool))),bNF_rel_fun(list(A),list(C),fun(list(B),bool),fun(list(D),bool),aa(fun(A,fun(C,bool)),fun(list(A),fun(list(C),bool)),list_all2(A,C),Sa),bNF_rel_fun(list(B),list(D),bool,bool,aa(fun(B,fun(D,bool)),fun(list(B),fun(list(D),bool)),list_all2(B,D),Sc),fequal(bool)))),list_all2(A,B)),list_all2(C,D))) ).

% list.rel_transfer
tff(fact_7997_list__update__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),fun(nat,fun(B,list(B)))),bool,aa(fun(list(A),fun(nat,fun(A,list(A)))),fun(fun(list(B),fun(nat,fun(B,list(B)))),bool),bNF_rel_fun(list(A),list(B),fun(nat,fun(A,list(A))),fun(nat,fun(B,list(B))),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),bNF_rel_fun(nat,nat,fun(A,list(A)),fun(B,list(B)),fequal(nat),bNF_rel_fun(A,B,list(A),list(B),A3,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3)))),list_update(A)),list_update(B))) ).

% list_update_transfer
tff(fact_7998_subseqs__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(list(B))),bool,aa(fun(list(A),list(list(A))),fun(fun(list(B),list(list(B))),bool),bNF_rel_fun(list(A),list(B),list(list(A)),list(list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(list(A),fun(list(B),bool)),fun(list(list(A)),fun(list(list(B)),bool)),list_all2(list(A),list(B)),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),subseqs(A)),subseqs(B))) ).

% subseqs_transfer
tff(fact_7999_rotate1__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3)),rotate1(A)),rotate1(B))) ).

% rotate1_transfer
tff(fact_8000_list_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),fequal(bool)),aTP_Lamp_akp(list(A),bool)),aTP_Lamp_akq(list(B),bool))) ).

% list.disc_transfer(1)
tff(fact_8001_list_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),fequal(bool)),aTP_Lamp_aey(list(A),bool)),aTP_Lamp_afk(list(B),bool))) ).

% list.disc_transfer(2)
tff(fact_8002_drop__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(nat,fun(list(B),list(B))),bool,aa(fun(nat,fun(list(A),list(A))),fun(fun(nat,fun(list(B),list(B))),bool),bNF_rel_fun(nat,nat,fun(list(A),list(A)),fun(list(B),list(B)),fequal(nat),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),drop(A)),drop(B))) ).

% drop_transfer
tff(fact_8003_take__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(nat,fun(list(B),list(B))),bool,aa(fun(nat,fun(list(A),list(A))),fun(fun(nat,fun(list(B),list(B))),bool),bNF_rel_fun(nat,nat,fun(list(A),list(A)),fun(list(B),list(B)),fequal(nat),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),take(A)),take(B))) ).

% take_transfer
tff(fact_8004_length__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),nat),bool,aa(fun(list(A),nat),fun(fun(list(B),nat),bool),bNF_rel_fun(list(A),list(B),nat,nat,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),fequal(nat)),size_size(list(A))),size_size(list(B)))) ).

% length_transfer
tff(fact_8005_rev__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),list(B)),bool,aa(fun(list(A),list(A)),fun(fun(list(B),list(B)),bool),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3)),rev(A)),rev(B))) ).

% rev_transfer
tff(fact_8006_list_Ocase__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,S2: fun(C,fun(D,bool)),R: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,fun(list(B),D)),fun(list(B),D))),bool,aa(fun(C,fun(fun(A,fun(list(A),C)),fun(list(A),C))),fun(fun(D,fun(fun(B,fun(list(B),D)),fun(list(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,fun(list(A),C)),fun(list(A),C)),fun(fun(B,fun(list(B),D)),fun(list(B),D)),S2,bNF_rel_fun(fun(A,fun(list(A),C)),fun(B,fun(list(B),D)),fun(list(A),C),fun(list(B),D),bNF_rel_fun(A,B,fun(list(A),C),fun(list(B),D),R,bNF_rel_fun(list(A),list(B),C,D,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),S2)),bNF_rel_fun(list(A),list(B),C,D,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R),S2))),case_list(C,A)),case_list(D,B))) ).

% list.case_transfer
tff(fact_8007_rotate__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(nat,fun(list(B),list(B))),bool,aa(fun(nat,fun(list(A),list(A))),fun(fun(nat,fun(list(B),list(B))),bool),bNF_rel_fun(nat,nat,fun(list(A),list(A)),fun(list(B),list(B)),fequal(nat),bNF_rel_fun(list(A),list(B),list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3))),rotate(A)),rotate(B))) ).

% rotate_transfer
tff(fact_8008_less__eq__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less_eq(nat)),ord_less_eq(nat))) ).

% less_eq_natural.rsp
tff(fact_8009_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R,one_one(A)),one_one(B)))
           => pp(aa(fun(bool,B),bool,aa(fun(bool,A),fun(fun(bool,B),bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),R),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).

% transfer_rule_of_bool
tff(fact_8010_less__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less(nat)),ord_less(nat))) ).

% less_natural.rsp
tff(fact_8011_less__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(int,int,fun(int,bool),fun(int,bool),fequal(int),bNF_rel_fun(int,int,bool,bool,fequal(int),fequal(bool))),ord_less(int)),ord_less(int))) ).

% less_integer.rsp
tff(fact_8012_maps__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,list(A))] : maps(B,A,F2,nil(B)) = nil(A) ).

% maps_simps(2)
tff(fact_8013_maps__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,list(A)),X: B,Xs: list(B)] : maps(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F2,X)),maps(B,A,F2,Xs)) ).

% maps_simps(1)
tff(fact_8014_concat__map__maps,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,list(A)),Xs: list(B)] : aa(list(list(A)),list(A),concat(A),aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),F2),Xs)) = maps(B,A,F2,Xs) ).

% concat_map_maps
tff(fact_8015_Real_Opositive_Otransfer,axiom,
    pp(aa(fun(real,bool),bool,aa(fun(fun(nat,rat),bool),fun(fun(real,bool),bool),bNF_rel_fun(fun(nat,rat),real,bool,bool,pcr_real,fequal(bool)),aTP_Lamp_ako(fun(nat,rat),bool)),positive2)) ).

% Real.positive.transfer
tff(fact_8016_plus__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_akr(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat))) ).

% plus_rat.transfer
tff(fact_8017_one__rat_Otransfer,axiom,
    pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),one_one(rat))) ).

% one_rat.transfer
tff(fact_8018_zero__rat_Otransfer,axiom,
    pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),zero_zero(rat))) ).

% zero_rat.transfer
tff(fact_8019_Fract_Otransfer,axiom,
    pp(aa(fun(int,fun(int,rat)),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),rat,fequal(int),pcr_rat)),aTP_Lamp_aks(int,fun(int,product_prod(int,int)))),fract)) ).

% Fract.transfer
tff(fact_8020_uminus__rat_Otransfer,axiom,
    pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_akt(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat))) ).

% uminus_rat.transfer
tff(fact_8021_times__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_aku(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat))) ).

% times_rat.transfer
tff(fact_8022_Rat_Opositive_Otransfer,axiom,
    pp(aa(fun(rat,bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(rat,bool),bool),bNF_rel_fun(product_prod(int,int),rat,bool,bool,pcr_rat,fequal(bool)),aTP_Lamp_akv(product_prod(int,int),bool)),positive)) ).

% Rat.positive.transfer
tff(fact_8023_inverse__rat_Otransfer,axiom,
    pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_akw(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat))) ).

% inverse_rat.transfer
tff(fact_8024_times__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ju(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).

% times_int.transfer
tff(fact_8025_minus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ke(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).

% minus_int.transfer
tff(fact_8026_zero__int_Otransfer,axiom,
    pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),zero_zero(int))) ).

% zero_int.transfer
tff(fact_8027_int__transfer,axiom,
    pp(aa(fun(nat,int),bool,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),bool),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_akx(nat,product_prod(nat,nat))),semiring_1_of_nat(int))) ).

% int_transfer
tff(fact_8028_uminus__int_Otransfer,axiom,
    pp(aa(fun(int,int),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),bool),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),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_jw(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).

% uminus_int.transfer
tff(fact_8029_one__int_Otransfer,axiom,
    pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),one_one(int))) ).

% one_int.transfer
tff(fact_8030_less__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_jy(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less(int))) ).

% less_int.transfer
tff(fact_8031_less__eq__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ka(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).

% less_eq_int.transfer
tff(fact_8032_plus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).

% plus_int.transfer
tff(fact_8033_times__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ju(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ju(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% times_int.rsp
tff(fact_8034_mono__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType] :
      ( ( order(B)
        & order(D)
        & order(C)
        & order(A) )
     => ! [A3: fun(A,fun(B,bool)),B3: fun(C,fun(D,bool))] :
          ( bi_total(A,B,A3)
         => ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),bool),bNF_rel_fun(A,B,fun(A,bool),fun(B,bool),A3,bNF_rel_fun(A,B,bool,bool,A3,fequal(bool))),ord_less_eq(A)),ord_less_eq(B)))
           => ( pp(aa(fun(D,fun(D,bool)),bool,aa(fun(C,fun(C,bool)),fun(fun(D,fun(D,bool)),bool),bNF_rel_fun(C,D,fun(C,bool),fun(D,bool),B3,bNF_rel_fun(C,D,bool,bool,B3,fequal(bool))),ord_less_eq(C)),ord_less_eq(D)))
             => pp(aa(fun(fun(B,D),bool),bool,aa(fun(fun(A,C),bool),fun(fun(fun(B,D),bool),bool),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A3,B3),fequal(bool)),order_mono(A,C)),order_mono(B,D))) ) ) ) ) ).

% mono_transfer
tff(fact_8035_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V2: nat] :
      ( pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),U),V2)))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).

% intrel_iff
tff(fact_8036_uminus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),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_jw(nat,fun(nat,product_prod(nat,nat))))),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_jw(nat,fun(nat,product_prod(nat,nat)))))) ).

% uminus_int.rsp
tff(fact_8037_zero__int_Orsp,axiom,
    pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat)))) ).

% zero_int.rsp
tff(fact_8038_list_Obi__total__rel,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,bool))] :
      ( bi_total(A,B,R)
     => bi_total(list(A),list(B),aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),R)) ) ).

% list.bi_total_rel
tff(fact_8039_one__int_Orsp,axiom,
    pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat)))) ).

% one_int.rsp
tff(fact_8040_less__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_jy(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_jy(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_int.rsp
tff(fact_8041_less__eq__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ka(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ka(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_eq_int.rsp
tff(fact_8042_plus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% plus_int.rsp
tff(fact_8043_minus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ke(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ke(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% minus_int.rsp
tff(fact_8044_monotone__parametric,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A3: fun(A,fun(B,bool)),B3: fun(C,fun(D,bool))] :
      ( bi_total(A,B,A3)
     => pp(aa(fun(fun(B,fun(B,bool)),fun(fun(D,fun(D,bool)),fun(fun(B,D),bool))),bool,aa(fun(fun(A,fun(A,bool)),fun(fun(C,fun(C,bool)),fun(fun(A,C),bool))),fun(fun(fun(B,fun(B,bool)),fun(fun(D,fun(D,bool)),fun(fun(B,D),bool))),bool),bNF_rel_fun(fun(A,fun(A,bool)),fun(B,fun(B,bool)),fun(fun(C,fun(C,bool)),fun(fun(A,C),bool)),fun(fun(D,fun(D,bool)),fun(fun(B,D),bool)),bNF_rel_fun(A,B,fun(A,bool),fun(B,bool),A3,bNF_rel_fun(A,B,bool,bool,A3,fequal(bool))),bNF_rel_fun(fun(C,fun(C,bool)),fun(D,fun(D,bool)),fun(fun(A,C),bool),fun(fun(B,D),bool),bNF_rel_fun(C,D,fun(C,bool),fun(D,bool),B3,bNF_rel_fun(C,D,bool,bool,B3,fequal(bool))),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A3,B3),fequal(bool)))),comple7038119648293358887notone(A,C)),comple7038119648293358887notone(B,D))) ) ).

% monotone_parametric
tff(fact_8045_fun__ord__parametric,axiom,
    ! [C: $tType,D: $tType,A: $tType,B: $tType,F: $tType,E: $tType,C3: fun(A,fun(B,bool)),A3: fun(C,fun(E,bool)),B3: fun(D,fun(F,bool))] :
      ( bi_total(A,B,C3)
     => pp(aa(fun(fun(E,fun(F,bool)),fun(fun(B,E),fun(fun(B,F),bool))),bool,aa(fun(fun(C,fun(D,bool)),fun(fun(A,C),fun(fun(A,D),bool))),fun(fun(fun(E,fun(F,bool)),fun(fun(B,E),fun(fun(B,F),bool))),bool),bNF_rel_fun(fun(C,fun(D,bool)),fun(E,fun(F,bool)),fun(fun(A,C),fun(fun(A,D),bool)),fun(fun(B,E),fun(fun(B,F),bool)),bNF_rel_fun(C,E,fun(D,bool),fun(F,bool),A3,bNF_rel_fun(D,F,bool,bool,B3,fequal(bool))),bNF_rel_fun(fun(A,C),fun(B,E),fun(fun(A,D),bool),fun(fun(B,F),bool),bNF_rel_fun(A,B,C,E,C3,A3),bNF_rel_fun(fun(A,D),fun(B,F),bool,bool,bNF_rel_fun(A,B,D,F,C3,B3),fequal(bool)))),partial_fun_ord(C,D,A)),partial_fun_ord(E,F,B))) ) ).

% fun_ord_parametric
tff(fact_8046_plus__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_akr(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_akr(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% plus_rat.rsp
tff(fact_8047_inverse__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_akw(product_prod(int,int),product_prod(int,int))),aTP_Lamp_akw(product_prod(int,int),product_prod(int,int)))) ).

% inverse_rat.rsp
tff(fact_8048_zero__rat_Orsp,axiom,
    pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)))) ).

% zero_rat.rsp
tff(fact_8049_one__rat_Orsp,axiom,
    pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)))) ).

% one_rat.rsp
tff(fact_8050_Fract_Orsp,axiom,
    pp(aa(fun(int,fun(int,product_prod(int,int))),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int)),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),product_prod(int,int),fequal(int),ratrel)),aTP_Lamp_aks(int,fun(int,product_prod(int,int)))),aTP_Lamp_aks(int,fun(int,product_prod(int,int))))) ).

% Fract.rsp
tff(fact_8051_uminus__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_akt(product_prod(int,int),product_prod(int,int))),aTP_Lamp_akt(product_prod(int,int),product_prod(int,int)))) ).

% uminus_rat.rsp
tff(fact_8052_times__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_aku(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_aku(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% times_rat.rsp
tff(fact_8053_Rat_Opositive_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),bool,bool,ratrel,fequal(bool)),aTP_Lamp_akv(product_prod(int,int),bool)),aTP_Lamp_akv(product_prod(int,int),bool))) ).

% Rat.positive.rsp
tff(fact_8054_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod(int,int),X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa2),Xa2))
     => ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa2)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_8055_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),X)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),X)),aa(product_prod(int,int),int,product_fst(int,int),X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_8056_one__rat__def,axiom,
    one_one(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).

% one_rat_def
tff(fact_8057_Fract_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(int,rat,aa(int,fun(int,rat),fract,Xa2),X) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),X),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa2),X))) ).

% Fract.abs_eq
tff(fact_8058_zero__rat__def,axiom,
    zero_zero(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).

% zero_rat_def
tff(fact_8059_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( aa(rat,rat,uminus_uminus(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(product_prod(int,int),int,product_snd(int,int),X))) ) ) ).

% uminus_rat.abs_eq
tff(fact_8060_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod(int,int),X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa2),Xa2))
     => ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
       => ( aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xa2)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa2)),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_8061_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( pp(aa(rat,bool,positive,aa(product_prod(int,int),rat,abs_Rat,X)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ).

% Rat.positive.abs_eq
tff(fact_8062_inverse__rat__def,axiom,
    inverse_inverse(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_akw(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_8063_uminus__rat__def,axiom,
    uminus_uminus(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_akt(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat_def
tff(fact_8064_plus__rat__def,axiom,
    plus_plus(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_akr(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_8065_times__rat__def,axiom,
    times_times(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_aku(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_8066_in__measures_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,F2: fun(A,nat),Fs: list(fun(A,nat))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y)))
        | ( ( aa(A,nat,F2,X) = aa(A,nat,F2,Y) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs))) ) ) ) ).

% in_measures(2)
tff(fact_8067_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R2)))
       => ( ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id(A))))
              | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id(A)))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A2),B2)) )
         => ( ( ( A2 = B2 )
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A2),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A2),B2)) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_8068_in__measures_I1_J,axiom,
    ! [A: $tType,X: A,Y: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,nil(fun(A,nat))))) ).

% in_measures(1)
tff(fact_8069_wo__rel_Omax2__among,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),bNF_We1388413361240627857o_max2(A,R2,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))) ) ) ) ).

% wo_rel.max2_among
tff(fact_8070_wf__measures,axiom,
    ! [A: $tType,Fs: list(fun(A,nat))] : wf(A,measures(A,Fs)) ).

% wf_measures
tff(fact_8071_wo__rel_Owell__order__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,bool),A2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( ! [X4: A] :
            ( ! [Y3: A] :
                ( ( ( Y3 != X4 )
                  & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4)),R2)) )
               => pp(aa(A,bool,P,Y3)) )
           => pp(aa(A,bool,P,X4)) )
       => pp(aa(A,bool,P,A2)) ) ) ).

% wo_rel.well_order_induct
tff(fact_8072_wo__rel_OTOTALS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),field2(A,R2)))
         => ! [Xa: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),field2(A,R2)))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Xa)),R2))
                | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X2)),R2)) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_8073_well__order__induct__imp,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,bool),A2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( ! [X4: A] :
            ( ! [Y3: A] :
                ( ( ( Y3 != X4 )
                  & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4)),R2)) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),field2(A,R2)))
                 => pp(aa(A,bool,P,Y3)) ) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),field2(A,R2)))
             => pp(aa(A,bool,P,X4)) ) )
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
         => pp(aa(A,bool,P,A2)) ) ) ) ).

% well_order_induct_imp
tff(fact_8074_wo__rel_Omax2__equals1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
         => ( ( bNF_We1388413361240627857o_max2(A,R2,A2,B2) = A2 )
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A2)),R2)) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_8075_wo__rel_Omax2__equals2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
         => ( ( bNF_We1388413361240627857o_max2(A,R2,A2,B2) = B2 )
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_8076_wo__rel_Omax2__greater,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),bNF_We1388413361240627857o_max2(A,R2,A2,B2))),R2))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R2,A2,B2))),R2)) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_8077_wo__rel_Omax2__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
         => ( bNF_We1388413361240627857o_max2(A,R2,A2,B2) = B2 ) )
        & ( ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
         => ( bNF_We1388413361240627857o_max2(A,R2,A2,B2) = A2 ) ) ) ) ).

% wo_rel.max2_def
tff(fact_8078_wo__rel_Ocases__Total,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R2)))
       => ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2))
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A2),B2)) )
         => ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A2)),R2))
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A2),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A2),B2)) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_8079_natLeq__on__wo__rel,axiom,
    ! [N: nat] : bNF_Wellorder_wo_rel(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ahh(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_wo_rel
tff(fact_8080_measures__less,axiom,
    ! [A: $tType,F2: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs)))) ) ).

% measures_less
tff(fact_8081_measures__lesseq,axiom,
    ! [A: $tType,F2: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F2,X)),aa(A,nat,F2,Y)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F2),Fs)))) ) ) ).

% measures_lesseq
tff(fact_8082_wo__rel_Omax2__greater__among,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R2)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),bNF_We1388413361240627857o_max2(A,R2,A2,B2))),R2))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R2,A2,B2))),R2))
            & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),bNF_We1388413361240627857o_max2(A,R2,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_8083_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R2)))
       => ( ( B3 != bot_bot(set(A)) )
         => ? [X_1: A] : bNF_We4791949203932849705sMinim(A,R2,B3,X_1) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_8084_wo__rel_Ominim__inField,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R2)))
       => ( ( B3 != bot_bot(set(A)) )
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),bNF_We6954850376910717587_minim(A,R2,B3)),field2(A,R2))) ) ) ) ).

% wo_rel.minim_inField
tff(fact_8085_wo__rel_Ominim__isMinim,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R2)))
       => ( ( B3 != bot_bot(set(A)) )
         => bNF_We4791949203932849705sMinim(A,R2,B3,bNF_We6954850376910717587_minim(A,R2,B3)) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_8086_wo__rel_OisMinim__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( bNF_We4791949203932849705sMinim(A,R2,A3,B2)
      <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3))
          & ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),X3)),R2)) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_8087_wo__rel_Oequals__minim,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A),A2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B3))
               => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B4)),R2)) )
           => ( A2 = bNF_We6954850376910717587_minim(A,R2,B3) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_8088_wo__rel_Ominim__least,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R2)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),B3))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R2,B3)),B2)),R2)) ) ) ) ).

% wo_rel.minim_least
tff(fact_8089_wo__rel_Ominim__in,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),field2(A,R2)))
       => ( ( B3 != bot_bot(set(A)) )
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),bNF_We6954850376910717587_minim(A,R2,B3)),B3)) ) ) ) ).

% wo_rel.minim_in
tff(fact_8090_null__transfer,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,aa(fun(A,fun(B,bool)),fun(list(A),fun(list(B),bool)),list_all2(A,B),A3),fequal(bool)),null(A)),null(B))) ).

% null_transfer
tff(fact_8091_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N)),transitive_rtrancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_8092_null__rec_I2_J,axiom,
    ! [B: $tType] : pp(aa(list(B),bool,null(B),nil(B))) ).

% null_rec(2)
tff(fact_8093_eq__Nil__null,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs = nil(A) )
    <=> pp(aa(list(A),bool,null(A),Xs)) ) ).

% eq_Nil_null
tff(fact_8094_null__rec_I1_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : ~ pp(aa(list(A),bool,null(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) ).

% null_rec(1)
tff(fact_8095_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N)),transitive_trancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% less_eq
tff(fact_8096_is__empty__set,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( is_empty(A,aa(list(A),set(A),set2(A),Xs))
    <=> pp(aa(list(A),bool,null(A),Xs)) ) ).

% is_empty_set
tff(fact_8097_VEBT_Osimps_I7_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),F22: fun(bool,fun(bool,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : vEBT_rec_VEBT(A,F1,F22,vEBT_Node(X11,X12,X13,X14)) = aa(A,A,aa(vEBT_VEBT,fun(A,A),aa(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)),aa(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)))),F1,X11),X12),aa(list(vEBT_VEBT),list(product_prod(vEBT_VEBT,A)),aa(fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),fun(list(vEBT_VEBT),list(product_prod(vEBT_VEBT,A))),map(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_aky(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),F1),F22)),X13)),X14),vEBT_rec_VEBT(A,F1,F22,X14)) ).

% VEBT.simps(7)
tff(fact_8098_VEBT_Osimps_I8_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),F22: fun(bool,fun(bool,A)),X21: bool,X222: bool] : vEBT_rec_VEBT(A,F1,F22,vEBT_Leaf(X21,X222)) = aa(bool,A,aa(bool,fun(bool,A),F22,X21),X222) ).

% VEBT.simps(8)
tff(fact_8099_Set_Ois__empty__def,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_empty(A,A3)
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% Set.is_empty_def
tff(fact_8100_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_8101_admissible__chfin,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,bool)] :
          ( ! [S6: set(A)] :
              ( comple1602240252501008431_chain(A,ord_less_eq(A),S6)
             => pp(aa(set(A),bool,finite_finite2(A),S6)) )
         => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P) ) ) ).

% admissible_chfin
tff(fact_8102_folding__idem_Ocomp__fun__idem,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),X: A] :
      ( finite_folding_idem(A,B,F2)
     => ( 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.comp_fun_idem
tff(fact_8103_ccpo_Oadmissible__def,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),P: fun(A,bool)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,P)
    <=> ! [A8: set(A)] :
          ( comple1602240252501008431_chain(A,Ord,A8)
         => ( ( A8 != bot_bot(set(A)) )
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A8))
                 => pp(aa(A,bool,P,X3)) )
             => pp(aa(A,bool,P,aa(set(A),A,Lub,A8))) ) ) ) ) ).

% ccpo.admissible_def
tff(fact_8104_ccpo_OadmissibleI,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool)),P: fun(A,bool),Lub: fun(set(A),A)] :
      ( ! [A5: set(A)] :
          ( comple1602240252501008431_chain(A,Ord,A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A5))
                 => pp(aa(A,bool,P,X2)) )
             => pp(aa(A,bool,P,aa(set(A),A,Lub,A5))) ) ) )
     => comple1908693960933563346ssible(A,Lub,Ord,P) ) ).

% ccpo.admissibleI
tff(fact_8105_ccpo_OadmissibleD,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),P: fun(A,bool),A3: set(A)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,P)
     => ( comple1602240252501008431_chain(A,Ord,A3)
       => ( ( A3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => pp(aa(A,bool,P,X4)) )
           => pp(aa(A,bool,P,aa(set(A),A,Lub,A3))) ) ) ) ) ).

% ccpo.admissibleD
tff(fact_8106_admissible__disj,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,bool),Q: fun(A,bool)] :
          ( 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,bool),fun(A,bool),aTP_Lamp_akz(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) ) ) ) ).

% admissible_disj
tff(fact_8107_fixp__induct,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,bool),F2: fun(A,A)] :
          ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
         => ( pp(aa(fun(A,A),bool,aa(fun(A,fun(A,bool)),fun(fun(A,A),bool),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(fun(A,A),bool)),comple7038119648293358887notone(A,A),ord_less_eq(A)),ord_less_eq(A)),F2))
           => ( pp(aa(A,bool,P,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A)))))
             => ( ! [X4: A] :
                    ( pp(aa(A,bool,P,X4))
                   => pp(aa(A,bool,P,aa(A,A,F2,X4))) )
               => pp(aa(A,bool,P,comple115746919287870866o_fixp(A,F2))) ) ) ) ) ) ).

% fixp_induct
tff(fact_8108_distinct__adj__append__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( distinct_adj(A,Xs)
        & distinct_adj(A,Ys)
        & ( ( Xs = nil(A) )
          | ( Ys = nil(A) )
          | ( last(A,Xs) != aa(list(A),A,hd(A),Ys) ) ) ) ) ).

% distinct_adj_append_iff
tff(fact_8109_distinct__adj__Cons__Cons,axiom,
    ! [B: $tType,X: B,Y: B,Xs: list(B)] :
      ( distinct_adj(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Xs)))
    <=> ( ( X != Y )
        & distinct_adj(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Xs)) ) ) ).

% distinct_adj_Cons_Cons
tff(fact_8110_distinct__adj__rev,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct_adj(A,aa(list(A),list(A),rev(A),Xs))
    <=> distinct_adj(A,Xs) ) ).

% distinct_adj_rev
tff(fact_8111_distinct__adj__ConsD,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( distinct_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))
     => distinct_adj(A,Xs) ) ).

% distinct_adj_ConsD
tff(fact_8112_distinct__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct_adj(A,Xs)
    <=> ( remdups_adj(A,Xs) = Xs ) ) ).

% distinct_adj_altdef
tff(fact_8113_distinct__adj__remdups__adj,axiom,
    ! [A: $tType,Xs: list(A)] : distinct_adj(A,remdups_adj(A,Xs)) ).

% distinct_adj_remdups_adj
tff(fact_8114_distinct__adj__appendD1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
     => distinct_adj(A,Xs) ) ).

% distinct_adj_appendD1
tff(fact_8115_distinct__adj__appendD2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
     => distinct_adj(A,Ys) ) ).

% distinct_adj_appendD2
tff(fact_8116_distinct__adj__Nil,axiom,
    ! [A: $tType] : distinct_adj(A,nil(A)) ).

% distinct_adj_Nil
tff(fact_8117_distinct__adj__singleton,axiom,
    ! [B: $tType,X: B] : distinct_adj(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B))) ).

% distinct_adj_singleton
tff(fact_8118_distinct__adj__mapD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( distinct_adj(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F2),Xs))
     => distinct_adj(B,Xs) ) ).

% distinct_adj_mapD
tff(fact_8119_distinct__adj__mapI,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B)] :
      ( distinct_adj(A,Xs)
     => ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xs))
       => distinct_adj(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs)) ) ) ).

% distinct_adj_mapI
tff(fact_8120_distinct__adj__map__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A)] :
      ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xs))
     => ( distinct_adj(B,aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F2),Xs))
      <=> distinct_adj(A,Xs) ) ) ).

% distinct_adj_map_iff
tff(fact_8121_distinct__adj__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( distinct_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))
    <=> ( ( Xs = nil(A) )
        | ( ( X != aa(list(A),A,hd(A),Xs) )
          & distinct_adj(A,Xs) ) ) ) ).

% distinct_adj_Cons
tff(fact_8122_fixp__unfold,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,aa(fun(A,fun(A,bool)),fun(fun(A,A),bool),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(fun(A,A),bool)),comple7038119648293358887notone(A,A),ord_less_eq(A)),ord_less_eq(A)),F2))
         => ( comple115746919287870866o_fixp(A,F2) = aa(A,A,F2,comple115746919287870866o_fixp(A,F2)) ) ) ) ).

% fixp_unfold
tff(fact_8123_fixp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),Z: A] :
          ( pp(aa(fun(A,A),bool,aa(fun(A,fun(A,bool)),fun(fun(A,A),bool),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(fun(A,A),bool)),comple7038119648293358887notone(A,A),ord_less_eq(A)),ord_less_eq(A)),F2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F2,Z)),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),comple115746919287870866o_fixp(A,F2)),Z)) ) ) ) ).

% fixp_lowerbound
tff(fact_8124_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_ala(A,fun(A,bool)),aTP_Lamp_alb(A,fun(A,bool)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_8125_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,bit_concat_bit(M,K,L)),N))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) )
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_8126_concat__bit__0,axiom,
    ! [K: int,L: int] : bit_concat_bit(zero_zero(nat),K,L) = L ).

% concat_bit_0
tff(fact_8127_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_concat_bit(N,K,L)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ).

% concat_bit_negative_iff
tff(fact_8128_ordering__top_Oextremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A2),Top)) ) ).

% ordering_top.extremum
tff(fact_8129_ordering__top_Oextremum__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Top),A2)) ) ).

% ordering_top.extremum_strict
tff(fact_8130_ordering__top_Oextremum__unique,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A2))
      <=> ( A2 = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_8131_ordering__top_Onot__eq__extremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( ( A2 != Top )
      <=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A2),Top)) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_8132_ordering__top_Oextremum__uniqueI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A2: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A2))
       => ( A2 = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_8133_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_jr(nat,fun(nat,bool)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_8134_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_8135_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_ak(nat,fun(nat,bool)),aTP_Lamp_aj(nat,fun(nat,bool)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_8136_span__singleton,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))) = aa(set(real),set(A),image2(real,A,aTP_Lamp_alc(A,fun(real,A),X)),top_top(set(real))) ) ).

% span_singleton
tff(fact_8137_span__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A)] : real_Vector_span(A,B2) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ald(set(A),fun(A,bool),B2)) ) ).

% span_explicit
tff(fact_8138_span__insert__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] : real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),S2)) = real_Vector_span(A,S2) ) ).

% span_insert_0
tff(fact_8139_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_8140_span__delete__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] : real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),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,S2) ) ).

% span_delete_0
tff(fact_8141_maximal__independent__subset__extend,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A),V: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),V))
         => ( ~ real_V358717886546972837endent(A,S2)
           => ~ ! [B8: set(A)] :
                  ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),B8))
                 => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),V))
                   => ( ~ real_V358717886546972837endent(A,B8)
                     => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V),real_Vector_span(A,B8))) ) ) ) ) ) ) ).

% maximal_independent_subset_extend
tff(fact_8142_spanning__subset__independent,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),A3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
         => ( ~ real_V358717886546972837endent(A,A3)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),real_Vector_span(A,B3)))
             => ( A3 = B3 ) ) ) ) ) ).

% spanning_subset_independent
tff(fact_8143_maximal__independent__subset,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A)] :
          ~ ! [B8: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),V))
             => ( ~ real_V358717886546972837endent(A,B8)
               => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V),real_Vector_span(A,B8))) ) ) ) ).

% maximal_independent_subset
tff(fact_8144_span__induct__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,S2: set(A),H: fun(A,bool)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,S2)))
         => ( pp(aa(A,bool,H,zero_zero(A)))
           => ( ! [C4: real,X4: A,Y2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                 => ( pp(aa(A,bool,H,Y2))
                   => pp(aa(A,bool,H,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,C4),X4)),Y2))) ) )
             => pp(aa(A,bool,H,X)) ) ) ) ) ).

% span_induct_alt
tff(fact_8145_span__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),real_Vector_span(A,S2))) ) ).

% span_0
tff(fact_8146_in__span__delete,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A,S2: set(A),B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_span(A,S2)))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))))) ) ) ) ).

% in_span_delete
tff(fact_8147_span__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A),T3: set(A)] :
          ( ( real_Vector_span(A,S2) = real_Vector_span(A,T3) )
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),real_Vector_span(A,T3)))
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),real_Vector_span(A,S2))) ) ) ) ).

% span_eq
tff(fact_8148_span__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),real_Vector_span(A,A3)),real_Vector_span(A,B3))) ) ) ).

% span_mono
tff(fact_8149_span__superset,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),real_Vector_span(A,S2))) ) ).

% span_superset
tff(fact_8150_span__finite,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( real_Vector_span(A,S2) = aa(set(fun(A,real)),set(A),image2(fun(A,real),A,aTP_Lamp_ale(set(A),fun(fun(A,real),A),S2)),top_top(set(fun(A,real)))) ) ) ) ).

% span_finite
tff(fact_8151_dependent__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [P: set(A)] :
          ( real_V358717886546972837endent(A,P)
        <=> ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),P))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),P),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))))) ) ) ) ).

% dependent_def
tff(fact_8152_span__image__scale,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A),C2: fun(A,real)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
               => ( aa(A,real,C2,X4) != zero_zero(real) ) )
           => ( real_Vector_span(A,aa(set(A),set(A),image2(A,A,aTP_Lamp_ajz(fun(A,real),fun(A,A),C2)),S2)) = real_Vector_span(A,S2) ) ) ) ) ).

% span_image_scale
tff(fact_8153_span__breakdown,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,S2: set(A),A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),S2))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_span(A,S2)))
           => ? [K2: real] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,K2),B2))),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))))) ) ) ) ).

% span_breakdown
tff(fact_8154_independent__span__bound,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [T3: set(A),S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),T3))
         => ( ~ real_V358717886546972837endent(A,S2)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),real_Vector_span(A,T3)))
             => ( pp(aa(set(A),bool,finite_finite2(A),S2))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),S2)),aa(set(A),nat,finite_card(A),T3))) ) ) ) ) ) ).

% independent_span_bound
tff(fact_8155_exchange__lemma,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [T3: set(A),S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),T3))
         => ( ~ real_V358717886546972837endent(A,S2)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),real_Vector_span(A,T3)))
             => ? [T13: set(A)] :
                  ( ( aa(set(A),nat,finite_card(A),T13) = aa(set(A),nat,finite_card(A),T3) )
                  & pp(aa(set(A),bool,finite_finite2(A),T13))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),T13))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T13),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),real_Vector_span(A,T13))) ) ) ) ) ) ).

% exchange_lemma
tff(fact_8156_span__explicit_H,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A)] : real_Vector_span(A,B2) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_alf(set(A),fun(A,bool),B2)) ) ).

% span_explicit'
tff(fact_8157_span__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] : real_Vector_span(A,B3) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_alg(set(A),fun(A,bool),B3)) ) ).

% span_alt
tff(fact_8158_representation__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A] :
          ( ( ( ~ real_V358717886546972837endent(A,Basis)
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),real_Vector_span(A,Basis))) )
           => ( real_V7696804695334737415tation(A,Basis,V2) = fChoice(fun(A,real),aa(A,fun(fun(A,real),bool),aTP_Lamp_alh(set(A),fun(A,fun(fun(A,real),bool)),Basis),V2)) ) )
          & ( ~ ( ~ real_V358717886546972837endent(A,Basis)
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),real_Vector_span(A,Basis))) )
           => ( real_V7696804695334737415tation(A,Basis,V2) = aTP_Lamp_ali(A,real) ) ) ) ) ).

% representation_def
tff(fact_8159_extend__basis__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] : real_V4986007116245087402_basis(A,B3) = fChoice(set(A),aTP_Lamp_alj(set(A),fun(set(A),bool),B3)) ) ).

% extend_basis_def
tff(fact_8160_finite__representation,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_alk(set(A),fun(A,fun(A,bool)),Basis),V2)))) ) ).

% finite_representation
tff(fact_8161_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_8162_representation__extend,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A,Basis2: set(A)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),real_Vector_span(A,Basis2)))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Basis2),Basis))
             => ( real_V7696804695334737415tation(A,Basis,V2) = real_V7696804695334737415tation(A,Basis2,V2) ) ) ) ) ) ).

% representation_extend
tff(fact_8163_extend__basis__superset,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] :
          ( ~ real_V358717886546972837endent(A,B3)
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),real_V4986007116245087402_basis(A,B3))) ) ) ).

% extend_basis_superset
tff(fact_8164_sum__representation__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A,B3: set(A)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),real_Vector_span(A,Basis)))
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Basis),B3))
               => ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aa(A,fun(A,A),aTP_Lamp_all(set(A),fun(A,fun(A,A)),Basis),V2)),B3) = V2 ) ) ) ) ) ) ).

% sum_representation_eq
tff(fact_8165_representation__eqI,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A,F2: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),real_Vector_span(A,Basis)))
           => ( ! [B4: A] :
                  ( ( aa(A,real,F2,B4) != zero_zero(real) )
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),Basis)) )
             => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),F2))))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),F2)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),F2))) = V2 )
                 => ( real_V7696804695334737415tation(A,Basis,V2) = F2 ) ) ) ) ) ) ) ).

% representation_eqI
tff(fact_8166_construct__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [B3: set(A),G: fun(A,B),V2: A] : real_V4425403222259421789struct(A,B,B3,G,V2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_alm(set(A),fun(fun(A,B),fun(A,fun(A,B))),B3),G),V2)),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aln(set(A),fun(A,fun(A,bool)),B3),V2))) ) ).

% construct_def
tff(fact_8167_dim__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A)] :
          ( ( ? [B9: set(A)] :
                ( ~ real_V358717886546972837endent(A,B9)
                & ( real_Vector_span(A,B9) = real_Vector_span(A,V) ) )
           => ( real_Vector_dim(A,V) = aa(set(A),nat,finite_card(A),fChoice(set(A),aTP_Lamp_alo(set(A),fun(set(A),bool),V))) ) )
          & ( ~ ? [B4: set(A)] :
                  ( ~ real_V358717886546972837endent(A,B4)
                  & ( real_Vector_span(A,B4) = real_Vector_span(A,V) ) )
           => ( real_Vector_dim(A,V) = zero_zero(nat) ) ) ) ) ).

% dim_def
tff(fact_8168_dim__le__card_H,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),real_Vector_dim(A,S)),aa(set(A),nat,finite_card(A),S))) ) ) ).

% dim_le_card'
tff(fact_8169_dim__unique,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),V: set(A),N: nat] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),V))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V),real_Vector_span(A,B3)))
           => ( ~ real_V358717886546972837endent(A,B3)
             => ( ( aa(set(A),nat,finite_card(A),B3) = N )
               => ( real_Vector_dim(A,V) = N ) ) ) ) ) ) ).

% dim_unique
tff(fact_8170_basis__exists,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A)] :
          ~ ! [B8: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),V))
             => ( ~ real_V358717886546972837endent(A,B8)
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V),real_Vector_span(A,B8)))
                 => ( aa(set(A),nat,finite_card(A),B8) != real_Vector_dim(A,V) ) ) ) ) ) ).

% basis_exists
tff(fact_8171_basis__card__eq__dim,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),V: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),V))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V),real_Vector_span(A,B3)))
           => ( ~ real_V358717886546972837endent(A,B3)
             => ( aa(set(A),nat,finite_card(A),B3) = real_Vector_dim(A,V) ) ) ) ) ) ).

% basis_card_eq_dim
tff(fact_8172_dim__le__card,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A),W3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V),real_Vector_span(A,W3)))
         => ( pp(aa(set(A),bool,finite_finite2(A),W3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),real_Vector_dim(A,V)),aa(set(A),nat,finite_card(A),W3))) ) ) ) ).

% dim_le_card
tff(fact_8173_span__card__ge__dim,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),V: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),V))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V),real_Vector_span(A,B3)))
           => ( pp(aa(set(A),bool,finite_finite2(A),B3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),real_Vector_dim(A,V)),aa(set(A),nat,finite_card(A),B3))) ) ) ) ) ).

% span_card_ge_dim
tff(fact_8174_construct__outside,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [B3: set(A),V2: A,F2: fun(A,B)] :
          ( ~ real_V358717886546972837endent(A,B3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),real_V4986007116245087402_basis(A,B3)),B3))))
           => ( real_V4425403222259421789struct(A,B,B3,F2,V2) = zero_zero(B) ) ) ) ) ).

% construct_outside
tff(fact_8175_pred__on_Onot__maxchain__Some,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,fun(A,bool)),C3: set(A)] :
      ( pp(aa(set(A),bool,pred_chain(A,A3,P),C3))
     => ( ~ pred_maxchain(A,A3,P,C3)
       => ( pp(aa(set(A),bool,pred_chain(A,A3,P),fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_alp(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A3),P),C3))))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),C3),fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_alp(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A3),P),C3)))) ) ) ) ).

% pred_on.not_maxchain_Some
tff(fact_8176_linear__indep__image__lemma,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),B3: set(A),X: A] :
          ( real_Vector_linear(A,B,F2)
         => ( pp(aa(set(A),bool,finite_finite2(A),B3))
           => ( ~ real_V358717886546972837endent(B,aa(set(A),set(B),image2(A,B,F2),B3))
             => ( inj_on(A,B,F2,B3)
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,B3)))
                 => ( ( aa(A,B,F2,X) = zero_zero(B) )
                   => ( X = zero_zero(A) ) ) ) ) ) ) ) ) ).

% linear_indep_image_lemma
tff(fact_8177_linear__eq__0__on__span,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F2: fun(A,B),B2: set(A),X: A] :
          ( real_Vector_linear(A,B,F2)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B2))
               => ( aa(A,B,F2,X4) = zero_zero(B) ) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,B2)))
             => ( aa(A,B,F2,X) = zero_zero(B) ) ) ) ) ) ).

% linear_eq_0_on_span
tff(fact_8178_subset_Omaxchain__imp__chain,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(set(A))] :
      ( pred_maxchain(set(A),A3,ord_less(set(A)),C3)
     => pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),C3)) ) ).

% subset.maxchain_imp_chain
tff(fact_8179_subset_Omaxchain__def,axiom,
    ! [A: $tType,A3: set(set(A)),C3: set(set(A))] :
      ( pred_maxchain(set(A),A3,ord_less(set(A)),C3)
    <=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),C3))
        & ~ ? [S10: set(set(A))] :
              ( pp(aa(set(set(A)),bool,pred_chain(set(A),A3,ord_less(set(A))),S10))
              & pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less(set(set(A))),C3),S10)) ) ) ) ).

% subset.maxchain_def
tff(fact_8180_subset_OHausdorff,axiom,
    ! [A: $tType,A3: set(set(A))] :
    ? [X_1: set(set(A))] : pred_maxchain(set(A),A3,ord_less(set(A)),X_1) ).

% subset.Hausdorff
tff(fact_8181_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_8182_module__hom__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => real_Vector_linear(A,B,aTP_Lamp_alq(A,B)) ) ).

% module_hom_zero
tff(fact_8183_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)))
          <=> ! [X3: A] :
                ( ( aa(A,B,F2,X3) = zero_zero(B) )
               => ( X3 = zero_zero(A) ) ) ) ) ) ).

% linear_injective_0
tff(fact_8184_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_akw(product_prod(int,int),product_prod(int,int)),Uu) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uu))) ).

% ATP.lambda_1
tff(fact_8185_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_pj(A,A),Uu) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Uu)),one_one(A)),Uu) ) ).

% ATP.lambda_2
tff(fact_8186_ATP_Olambda__3,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_iw(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).

% ATP.lambda_3
tff(fact_8187_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_mc(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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_4
tff(fact_8188_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aik(set(A),bool),Uu))
    <=> ( ( Uu != bot_bot(set(A)) )
        & countable_countable(A,Uu) ) ) ).

% ATP.lambda_5
tff(fact_8189_ATP_Olambda__6,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_akt(product_prod(int,int),product_prod(int,int)),Uu) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(product_prod(int,int),int,product_snd(int,int),Uu)) ).

% ATP.lambda_6
tff(fact_8190_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( pp(aa(nat,bool,aTP_Lamp_kf(nat,bool),Uu))
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_7
tff(fact_8191_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_jm(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_8
tff(fact_8192_ATP_Olambda__9,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_jk(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).

% ATP.lambda_9
tff(fact_8193_ATP_Olambda__10,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_jl(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).

% ATP.lambda_10
tff(fact_8194_ATP_Olambda__11,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_cw(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_11
tff(fact_8195_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_akx(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uu),zero_zero(nat)) ).

% ATP.lambda_12
tff(fact_8196_ATP_Olambda__13,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_adc(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_13
tff(fact_8197_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_lp(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_14
tff(fact_8198_ATP_Olambda__15,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_akq(list(B),bool),Uu))
    <=> ( Uu = nil(B) ) ) ).

% ATP.lambda_15
tff(fact_8199_ATP_Olambda__16,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_akp(list(A),bool),Uu))
    <=> ( Uu = nil(A) ) ) ).

% ATP.lambda_16
tff(fact_8200_ATP_Olambda__17,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_akv(product_prod(int,int),bool),Uu))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))) ) ).

% ATP.lambda_17
tff(fact_8201_ATP_Olambda__18,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_qn(nat,A),Uu) = divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_18
tff(fact_8202_ATP_Olambda__19,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_afj(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_19
tff(fact_8203_ATP_Olambda__20,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_zs(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),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).

% ATP.lambda_20
tff(fact_8204_ATP_Olambda__21,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_afk(list(B),bool),Uu))
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_21
tff(fact_8205_ATP_Olambda__22,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aey(list(A),bool),Uu))
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_22
tff(fact_8206_ATP_Olambda__23,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_aco(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = aa(fun(A,fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(A,C),product_prod(A,product_prod(B,C))),product_case_prod(A,C,product_prod(A,product_prod(B,C))),aTP_Lamp_acn(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_23
tff(fact_8207_ATP_Olambda__24,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B))] : aa(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))),aTP_Lamp_abp(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Uu) = aa(fun(A,fun(B,fun(A,option(B)))),fun(product_prod(A,B),fun(A,option(B))),product_case_prod(A,B,fun(A,option(B))),aTP_Lamp_abo(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).

% ATP.lambda_24
tff(fact_8208_ATP_Olambda__25,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_wg(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_wf(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).

% ATP.lambda_25
tff(fact_8209_ATP_Olambda__26,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_nk(real,real),Uu) = suminf(real,aTP_Lamp_nj(real,fun(nat,real),Uu)) ).

% ATP.lambda_26
tff(fact_8210_ATP_Olambda__27,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_uu(nat,set(nat)),Uu) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_bd(nat,fun(nat,bool),Uu)) ).

% ATP.lambda_27
tff(fact_8211_ATP_Olambda__28,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_xe(fun(A,B),set(product_prod(A,B))),Uu) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_xd(fun(A,B),fun(A,fun(B,bool)),Uu))) ).

% ATP.lambda_28
tff(fact_8212_ATP_Olambda__29,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(real,real)),aTP_Lamp_vu(real,filter(product_prod(real,real))),Uu) = principal(product_prod(real,real),aa(fun(product_prod(real,real),bool),set(product_prod(real,real)),collect(product_prod(real,real)),aa(fun(real,fun(real,bool)),fun(product_prod(real,real),bool),product_case_prod(real,real,bool),aTP_Lamp_vt(real,fun(real,fun(real,bool)),Uu)))) ).

% ATP.lambda_29
tff(fact_8213_ATP_Olambda__30,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(complex,complex)),aTP_Lamp_vs(real,filter(product_prod(complex,complex))),Uu) = principal(product_prod(complex,complex),aa(fun(product_prod(complex,complex),bool),set(product_prod(complex,complex)),collect(product_prod(complex,complex)),aa(fun(complex,fun(complex,bool)),fun(product_prod(complex,complex),bool),product_case_prod(complex,complex,bool),aTP_Lamp_vr(real,fun(complex,fun(complex,bool)),Uu)))) ).

% ATP.lambda_30
tff(fact_8214_ATP_Olambda__31,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real] : aa(real,filter(product_prod(A,A)),aTP_Lamp_ve(real,filter(product_prod(A,A))),Uu) = principal(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_vd(real,fun(A,fun(A,bool)),Uu)))) ) ).

% ATP.lambda_31
tff(fact_8215_ATP_Olambda__32,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_qj(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_32
tff(fact_8216_ATP_Olambda__33,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_afi(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_33
tff(fact_8217_ATP_Olambda__34,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_afh(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_34
tff(fact_8218_ATP_Olambda__35,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_abu(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_35
tff(fact_8219_ATP_Olambda__36,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_aaq(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).

% ATP.lambda_36
tff(fact_8220_ATP_Olambda__37,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hp(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).

% ATP.lambda_37
tff(fact_8221_ATP_Olambda__38,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hq(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_38
tff(fact_8222_ATP_Olambda__39,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bj(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_39
tff(fact_8223_ATP_Olambda__40,axiom,
    ! [Uu: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aTP_Lamp_ako(fun(nat,rat),bool),Uu))
    <=> ? [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
          & ? [K3: nat] :
            ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N3))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,Uu,N3))) ) ) ) ).

% ATP.lambda_40
tff(fact_8224_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yj(product_prod(A,A),bool),Uu))
        <=> ? [X3: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ) ).

% ATP.lambda_41
tff(fact_8225_ATP_Olambda__42,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aTP_Lamp_aiq(product_prod(A,A),bool),Uu))
    <=> ? [X3: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).

% ATP.lambda_42
tff(fact_8226_ATP_Olambda__43,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_abz(real,bool),Uu))
    <=> ? [I4: int,N3: nat] :
          ( ( Uu = divide_divide(real,aa(int,real,ring_1_of_int(real),I4),aa(nat,real,semiring_1_of_nat(real),N3)) )
          & ( N3 != zero_zero(nat) ) ) ) ).

% ATP.lambda_43
tff(fact_8227_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_ym(product_prod(A,A),bool),Uu))
        <=> ? [X3: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4)) ) ) ) ).

% ATP.lambda_44
tff(fact_8228_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yl(product_prod(A,A),bool),Uu))
        <=> ? [X3: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X3)) ) ) ) ).

% ATP.lambda_45
tff(fact_8229_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yn(product_prod(A,A),bool),Uu))
        <=> ? [X3: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4)) ) ) ) ).

% ATP.lambda_46
tff(fact_8230_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yo(product_prod(A,A),bool),Uu))
        <=> ? [X3: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X3)) ) ) ) ).

% ATP.lambda_47
tff(fact_8231_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yg(product_prod(A,A),bool),Uu))
        <=> ? [X3: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4) )
              & ( X3 != Y4 ) ) ) ) ).

% ATP.lambda_48
tff(fact_8232_ATP_Olambda__49,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_aaw(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_49
tff(fact_8233_ATP_Olambda__50,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_aba(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_aay(nat,fun(num,option(num)),Uua),aTP_Lamp_aaz(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_50
tff(fact_8234_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fi(A,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_51
tff(fact_8235_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dz(nat,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ).

% ATP.lambda_52
tff(fact_8236_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fo(A,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_53
tff(fact_8237_ATP_Olambda__54,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: product_prod(A,C),Uua: product_prod(C,B)] : aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_acr(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = if(list(product_prod(A,B)),aa(C,bool,aa(C,fun(C,bool),fequal(C),aa(product_prod(A,C),C,product_snd(A,C),Uu)),aa(product_prod(C,B),C,product_fst(C,B),Uua)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).

% ATP.lambda_54
tff(fact_8238_ATP_Olambda__55,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_iq(code_integer,fun(code_integer,int)),Uu),Uua) = if(int,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),code_int_of_integer(Uu)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),code_int_of_integer(Uu))),one_one(int))) ).

% ATP.lambda_55
tff(fact_8239_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ct(nat,fun(nat,A)),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uua),zero_zero(nat)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),Uu))),one_one(A))) ) ).

% ATP.lambda_56
tff(fact_8240_ATP_Olambda__57,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_it(code_integer,fun(code_integer,nat)),Uu),Uua) = if(nat,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_integer,nat,code_nat_of_integer,Uu)),aa(code_integer,nat,code_nat_of_integer,Uu)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_integer,nat,code_nat_of_integer,Uu)),aa(code_integer,nat,code_nat_of_integer,Uu))),one_one(nat))) ).

% ATP.lambda_57
tff(fact_8241_ATP_Olambda__58,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ho(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uu),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_58
tff(fact_8242_ATP_Olambda__59,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aks(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Uu),Uua)) ).

% ATP.lambda_59
tff(fact_8243_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dy(nat,fun(nat,A),Uu),Uua) = if(A,aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ).

% ATP.lambda_60
tff(fact_8244_ATP_Olambda__61,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_wn(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_61
tff(fact_8245_ATP_Olambda__62,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_ml(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_62
tff(fact_8246_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_ajh(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_63
tff(fact_8247_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_wk(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_64
tff(fact_8248_ATP_Olambda__65,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aja(list(list(A)),fun(list(A),bool),Uu),Uua))
    <=> pp(aa(list(list(A)),bool,aa(list(A),fun(list(list(A)),bool),aa(fun(A,fun(list(A),bool)),fun(list(A),fun(list(list(A)),bool)),list_all2(A,list(A)),aTP_Lamp_aiz(A,fun(list(A),bool))),Uua),Uu)) ) ).

% ATP.lambda_65
tff(fact_8249_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_abb(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_aba(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_66
tff(fact_8250_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_aay(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_aaq(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_67
tff(fact_8251_ATP_Olambda__68,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_aax(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_aaq(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_68
tff(fact_8252_ATP_Olambda__69,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_aje(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_69
tff(fact_8253_ATP_Olambda__70,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_aea(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),aa(fun(nat,A),fun(list(nat),list(A)),map(nat,A),aa(nat,fun(nat,A),aTP_Lamp_adz(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_70
tff(fact_8254_ATP_Olambda__71,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hv(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_71
tff(fact_8255_ATP_Olambda__72,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_dk(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_dj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_72
tff(fact_8256_ATP_Olambda__73,axiom,
    ! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_akr(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_73
tff(fact_8257_ATP_Olambda__74,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fc(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_74
tff(fact_8258_ATP_Olambda__75,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_eb(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua))) ) ).

% ATP.lambda_75
tff(fact_8259_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_fq(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Uua),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_76
tff(fact_8260_ATP_Olambda__77,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,nat)] :
      ( pp(aa(fun(A,nat),bool,aTP_Lamp_ahz(set(A),fun(fun(A,nat),bool),Uu),Uua))
    <=> ( ( pp(aa(set(A),bool,finite_finite2(A),Uu))
         => bij_betw(A,nat,Uua,Uu,set_ord_lessThan(nat,aa(set(A),nat,finite_card(A),Uu))) )
        & ( ~ pp(aa(set(A),bool,finite_finite2(A),Uu))
         => bij_betw(A,nat,Uua,Uu,top_top(set(nat))) ) ) ) ).

% ATP.lambda_77
tff(fact_8261_ATP_Olambda__78,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_ahn(complex,fun(real,bool),Uu),Uua))
    <=> ( ( aa(complex,complex,sgn_sgn(complex),Uu) = cis(Uua) )
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),pi)) ) ) ).

% ATP.lambda_78
tff(fact_8262_ATP_Olambda__79,axiom,
    ! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_aku(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_79
tff(fact_8263_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_gh(real,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Uu),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_80
tff(fact_8264_ATP_Olambda__81,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_gl(rat,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu))
        & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_81
tff(fact_8265_ATP_Olambda__82,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_nj(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat))))) ).

% ATP.lambda_82
tff(fact_8266_ATP_Olambda__83,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_nl(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% ATP.lambda_83
tff(fact_8267_ATP_Olambda__84,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_qc(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_84
tff(fact_8268_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ea(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua))) ) ).

% ATP.lambda_85
tff(fact_8269_ATP_Olambda__86,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_aia(rat,fun(product_prod(int,int),bool),Uu),Uua))
    <=> ( ( Uu = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uua)) )
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Uua)))
        & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) ) ) ).

% ATP.lambda_86
tff(fact_8270_ATP_Olambda__87,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_ajm(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)),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_87
tff(fact_8271_ATP_Olambda__88,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_eq(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,Uu,Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uu,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_88
tff(fact_8272_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_alj(set(A),fun(set(A),bool),Uu),Uua))
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uu),Uua))
            & ~ real_V358717886546972837endent(A,Uua)
            & ( real_Vector_span(A,Uua) = top_top(set(A)) ) ) ) ) ).

% ATP.lambda_89
tff(fact_8273_ATP_Olambda__90,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_jr(nat,fun(nat,bool)),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_90
tff(fact_8274_ATP_Olambda__91,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_ix(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_91
tff(fact_8275_ATP_Olambda__92,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_mp(set(option(A)),fun(option(A),bool),Uu),Uua))
    <=> ( pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),Uua),Uu))
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_92
tff(fact_8276_ATP_Olambda__93,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_zm(set(set(A)),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Uua),Uu))
        & ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),Uu))
           => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),X3)) ) ) ) ).

% ATP.lambda_93
tff(fact_8277_ATP_Olambda__94,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_akn(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),Uu),Uua))
        & ! [A7: A,B7: A,C5: A] :
            ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7)),Uua))
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C5)),Uu)) )
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7)),Uu)) ) ) ) ).

% ATP.lambda_94
tff(fact_8278_ATP_Olambda__95,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ey(real,fun(nat,real),Uu),Uua) = divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_95
tff(fact_8279_ATP_Olambda__96,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_jq(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu))
        & pp(aa(set(A),bool,finite_finite2(A),Uua)) ) ) ).

% ATP.lambda_96
tff(fact_8280_ATP_Olambda__97,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_np(set(A),fun(set(A),bool)),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uu),Uua))
        & pp(aa(set(A),bool,finite_finite2(A),Uua)) ) ) ).

% ATP.lambda_97
tff(fact_8281_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_wr(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uua)),Uua) ).

% ATP.lambda_98
tff(fact_8282_ATP_Olambda__99,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_zu(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),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_99
tff(fact_8283_ATP_Olambda__100,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_agr(A,fun(B,set(product_prod(A,B))),Uu),Uua) = 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),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_100
tff(fact_8284_ATP_Olambda__101,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa(complex,bool,aTP_Lamp_cs(nat,fun(complex,bool),Uu),Uua))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_101
tff(fact_8285_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aw(nat,fun(A,bool),Uu),Uua))
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_102
tff(fact_8286_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_xt(A,fun(A,bool),Uu),Uua))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu)) ) ) ) ).

% ATP.lambda_103
tff(fact_8287_ATP_Olambda__104,axiom,
    ! [A: $tType,Uu: fun(set(A),bool),Uua: set(A)] :
      ( pp(aa(set(A),bool,aa(fun(set(A),bool),fun(set(A),bool),aTP_Lamp_yk(fun(set(A),bool),fun(set(A),bool)),Uu),Uua))
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A8: set(A),A7: A] :
            ( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),A8) )
            & pp(aa(set(A),bool,Uu,A8)) ) ) ) ).

% ATP.lambda_104
tff(fact_8288_ATP_Olambda__105,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_aid(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_105
tff(fact_8289_ATP_Olambda__106,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_wi(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ) ).

% ATP.lambda_106
tff(fact_8290_ATP_Olambda__107,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_wt(nat,fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu))) ) ) ).

% ATP.lambda_107
tff(fact_8291_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_qp(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_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_cl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_109
tff(fact_8293_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gr(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_110
tff(fact_8294_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hd(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_111
tff(fact_8295_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_de(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_112
tff(fact_8296_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_113
tff(fact_8297_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( division_ring(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,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_114
tff(fact_8298_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fu(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_115
tff(fact_8299_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cn(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_116
tff(fact_8300_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_117
tff(fact_8301_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_qo(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_118
tff(fact_8302_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_119
tff(fact_8303_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ajv(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y4: A] :
                ( pp(aa(A,bool,Uu,Y4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y4)) ) ) ) ) ).

% ATP.lambda_120
tff(fact_8304_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_te(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y4: A] :
                ( pp(aa(A,bool,Uu,Y4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Uua)) ) ) ) ) ).

% ATP.lambda_121
tff(fact_8305_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,A,aTP_Lamp_ajz(fun(A,real),fun(A,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(A,real,Uu,Uua)),Uua) ) ).

% ATP.lambda_122
tff(fact_8306_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ajf(fun(A,A),fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,Uu,Uua)),Uua)) ) ) ).

% ATP.lambda_123
tff(fact_8307_ATP_Olambda__124,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_rv(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,Uu,Uua)),zero_zero(real))) ) ).

% ATP.lambda_124
tff(fact_8308_ATP_Olambda__125,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_act(fun(B,A),fun(B,list(A)),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)),nil(A)) ).

% ATP.lambda_125
tff(fact_8309_ATP_Olambda__126,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_kt(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_126
tff(fact_8310_ATP_Olambda__127,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_cj(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = zero_zero(A) ) ) ) ).

% ATP.lambda_127
tff(fact_8311_ATP_Olambda__128,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_ha(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).

% ATP.lambda_128
tff(fact_8312_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: fun(A,real)] : aa(fun(A,real),A,aTP_Lamp_ale(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_ajz(fun(A,real),fun(A,A),Uua)),Uu) ) ).

% ATP.lambda_129
tff(fact_8313_ATP_Olambda__130,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_acx(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),aa(fun(A,list(A)),fun(list(A),list(list(A))),map(A,list(A)),aa(list(A),fun(A,list(A)),aTP_Lamp_acw(list(A),fun(A,list(A))),Uua)),Uu) ).

% ATP.lambda_130
tff(fact_8314_ATP_Olambda__131,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_qv(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_131
tff(fact_8315_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_alo(set(A),fun(set(A),bool),Uu),Uua))
        <=> ( ~ real_V358717886546972837endent(A,Uua)
            & ( real_Vector_span(A,Uua) = real_Vector_span(A,Uu) ) ) ) ) ).

% ATP.lambda_132
tff(fact_8316_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_da(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_133
tff(fact_8317_ATP_Olambda__134,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_mn(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_134
tff(fact_8318_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_qy(A,fun(nat,A),Uu),Uua) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_135
tff(fact_8319_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_qx(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_136
tff(fact_8320_ATP_Olambda__137,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fn(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_137
tff(fact_8321_ATP_Olambda__138,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fp(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_138
tff(fact_8322_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_tk(A,fun(set(A),bool),Uu),Uua))
        <=> ( topolo1002775350975398744n_open(A,Uua)
            & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uu),Uua)) ) ) ) ).

% ATP.lambda_139
tff(fact_8323_ATP_Olambda__140,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_za(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uua))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).

% ATP.lambda_140
tff(fact_8324_ATP_Olambda__141,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: list(product_prod(A,B))] :
      ( pp(aa(list(product_prod(A,B)),bool,aTP_Lamp_ait(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).

% ATP.lambda_141
tff(fact_8325_ATP_Olambda__142,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_acv(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Uua)),aa(list(list(A)),list(list(A)),product_lists(A),Uu)) ).

% ATP.lambda_142
tff(fact_8326_ATP_Olambda__143,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_zn(list(A),fun(list(A),bool)),Uu),Uua))
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_143
tff(fact_8327_ATP_Olambda__144,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aib(set(A),fun(list(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu)) ) ).

% ATP.lambda_144
tff(fact_8328_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_er(nat,fun(nat,A),Uu),Uua) = gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua),Uu) ) ).

% ATP.lambda_145
tff(fact_8329_ATP_Olambda__146,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_acu(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(list(B),list(product_prod(A,B))),map(B,product_prod(A,B)),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua)),Uu) ).

% ATP.lambda_146
tff(fact_8330_ATP_Olambda__147,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_afe(set(nat),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)),Uu)) ) ).

% ATP.lambda_147
tff(fact_8331_ATP_Olambda__148,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_adf(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,Uua)),Uu)) ) ).

% ATP.lambda_148
tff(fact_8332_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_bk(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_149
tff(fact_8333_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_bi(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_150
tff(fact_8334_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gk(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),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_151
tff(fact_8335_ATP_Olambda__152,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_zp(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_152
tff(fact_8336_ATP_Olambda__153,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_ajl(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = 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_153
tff(fact_8337_ATP_Olambda__154,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_ahd(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_154
tff(fact_8338_ATP_Olambda__155,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_fy(set(A),fun(A,bool),Uu),Uua))
    <=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_155
tff(fact_8339_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_qs(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_156
tff(fact_8340_ATP_Olambda__157,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_adw(fun(nat,A),fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu,Uua)) ).

% ATP.lambda_157
tff(fact_8341_ATP_Olambda__158,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_xc(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).

% ATP.lambda_158
tff(fact_8342_ATP_Olambda__159,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_js(A,fun(nat,A),Uu),Uua) = bit_se4730199178511100633sh_bit(A,Uua,aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_159
tff(fact_8343_ATP_Olambda__160,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_abt(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu,Uua))) ).

% ATP.lambda_160
tff(fact_8344_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_qt(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_161
tff(fact_8345_ATP_Olambda__162,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_wq(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ).

% ATP.lambda_162
tff(fact_8346_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_qi(A,fun(nat,A),Uu),Uua) = divide_divide(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_163
tff(fact_8347_ATP_Olambda__164,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aef(nat,fun(list(A),bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua))) ) ).

% ATP.lambda_164
tff(fact_8348_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gz(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_165
tff(fact_8349_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gy(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_166
tff(fact_8350_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gu(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_167
tff(fact_8351_ATP_Olambda__168,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_aiz(A,fun(list(A),bool)),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uu),aa(list(A),set(A),set2(A),Uua))) ) ).

% ATP.lambda_168
tff(fact_8352_ATP_Olambda__169,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aep(list(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),aa(list(A),set(A),set2(A),Uu))) ) ).

% ATP.lambda_169
tff(fact_8353_ATP_Olambda__170,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_afz(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).

% ATP.lambda_170
tff(fact_8354_ATP_Olambda__171,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_akh(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua = last(A,Uu) ) ) ).

% ATP.lambda_171
tff(fact_8355_ATP_Olambda__172,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A,Uua: real] : aa(real,A,aTP_Lamp_alc(A,fun(real,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_8356_ATP_Olambda__173,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ag(set(A),fun(set(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ).

% ATP.lambda_173
tff(fact_8357_ATP_Olambda__174,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ak(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uu)) ) ).

% ATP.lambda_174
tff(fact_8358_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_yw(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_175
tff(fact_8359_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ala(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_176
tff(fact_8360_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_rx(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_177
tff(fact_8361_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_df(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_178
tff(fact_8362_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mk(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_179
tff(fact_8363_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ko(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_180
tff(fact_8364_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wu(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_181
tff(fact_8365_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).

% ATP.lambda_182
tff(fact_8366_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_yx(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_183
tff(fact_8367_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_rw(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_184
tff(fact_8368_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_alb(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_185
tff(fact_8369_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_akj(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_186
tff(fact_8370_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_cx(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_187
tff(fact_8371_ATP_Olambda__188,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_qh(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_188
tff(fact_8372_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ld(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_8373_ATP_Olambda__190,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kh(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_190
tff(fact_8374_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_mu(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_191
tff(fact_8375_ATP_Olambda__192,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_lw(set(A),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),Uu) ).

% ATP.lambda_192
tff(fact_8376_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_adu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_193
tff(fact_8377_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wo(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_194
tff(fact_8378_ATP_Olambda__195,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_mw(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_195
tff(fact_8379_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_bd(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uua),Uu)) ) ).

% ATP.lambda_196
tff(fact_8380_ATP_Olambda__197,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_bc(int,fun(int,bool),Uu),Uua))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uua),Uu)) ) ).

% ATP.lambda_197
tff(fact_8381_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_bb(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),Uu)) ) ) ).

% ATP.lambda_198
tff(fact_8382_ATP_Olambda__199,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_jw(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uu) ).

% ATP.lambda_199
tff(fact_8383_ATP_Olambda__200,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_jn(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_200
tff(fact_8384_ATP_Olambda__201,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ady(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),Uu) ).

% ATP.lambda_201
tff(fact_8385_ATP_Olambda__202,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_jo(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).

% ATP.lambda_202
tff(fact_8386_ATP_Olambda__203,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_acw(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu) ).

% ATP.lambda_203
tff(fact_8387_ATP_Olambda__204,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : aa(A,A,aTP_Lamp_xh(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ).

% ATP.lambda_204
tff(fact_8388_ATP_Olambda__205,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_qm(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_205
tff(fact_8389_ATP_Olambda__206,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_tf(set(B),fun(B,bool),Uu),Uua))
    <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uua),Uu)) ) ).

% ATP.lambda_206
tff(fact_8390_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_wz(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_207
tff(fact_8391_ATP_Olambda__208,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ajr(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_208
tff(fact_8392_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aju(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_209
tff(fact_8393_ATP_Olambda__210,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_a(set(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_210
tff(fact_8394_ATP_Olambda__211,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_wh(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_211
tff(fact_8395_ATP_Olambda__212,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_adj(list(A),fun(nat,list(A)),Uu),Uua) = aa(list(A),list(A),aa(nat,fun(list(A),list(A)),drop(A),Uua),Uu) ).

% ATP.lambda_212
tff(fact_8396_ATP_Olambda__213,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_aee(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_213
tff(fact_8397_ATP_Olambda__214,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_bp(A,fun(A,bool),Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_214
tff(fact_8398_ATP_Olambda__215,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ado(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_215
tff(fact_8399_ATP_Olambda__216,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_adp(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Uua),nil(list(A))) ).

% ATP.lambda_216
tff(fact_8400_ATP_Olambda__217,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_sw(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_217
tff(fact_8401_ATP_Olambda__218,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_sg(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_218
tff(fact_8402_ATP_Olambda__219,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_ahi(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_219
tff(fact_8403_ATP_Olambda__220,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_re(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,divide_divide(A,one_one(A),Uua)) ) ).

% ATP.lambda_220
tff(fact_8404_ATP_Olambda__221,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_afc(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)))) ) ).

% ATP.lambda_221
tff(fact_8405_ATP_Olambda__222,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_afd(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_222
tff(fact_8406_ATP_Olambda__223,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ajq(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_223
tff(fact_8407_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_eh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_224
tff(fact_8408_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ei(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_225
tff(fact_8409_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hc(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_226
tff(fact_8410_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ck(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_227
tff(fact_8411_ATP_Olambda__228,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_adx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_228
tff(fact_8412_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_abn(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_229
tff(fact_8413_ATP_Olambda__230,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_zt(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_230
tff(fact_8414_ATP_Olambda__231,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_zz(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_231
tff(fact_8415_ATP_Olambda__232,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_aaa(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_232
tff(fact_8416_ATP_Olambda__233,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_aaz(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_233
tff(fact_8417_ATP_Olambda__234,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_aau(num,fun(nat,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_234
tff(fact_8418_ATP_Olambda__235,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_aec(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_aeb(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_235
tff(fact_8419_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ke(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_kd(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_236
tff(fact_8420_ATP_Olambda__237,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_kb(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_237
tff(fact_8421_ATP_Olambda__238,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ka(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_jz(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_238
tff(fact_8422_ATP_Olambda__239,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_jy(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_jx(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_239
tff(fact_8423_ATP_Olambda__240,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ju(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_jt(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_240
tff(fact_8424_ATP_Olambda__241,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_no(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_nn(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_241
tff(fact_8425_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_my(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_gi(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_242
tff(fact_8426_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A] : aa(A,set(A),aTP_Lamp_rq(real,fun(A,set(A)),Uu),Uua) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_rp(real,fun(A,fun(A,bool)),Uu),Uua)) ) ).

% ATP.lambda_243
tff(fact_8427_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_ahk(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),aa(num,num,bit0,one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua)),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_244
tff(fact_8428_ATP_Olambda__245,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_pp(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_245
tff(fact_8429_ATP_Olambda__246,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aka(fun(A,real),fun(A,bool),Uu),Uua))
        <=> ( aa(A,real,Uu,Uua) != zero_zero(real) ) ) ) ).

% ATP.lambda_246
tff(fact_8430_ATP_Olambda__247,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_abr(fun(A,option(B)),fun(A,bool),Uu),Uua))
    <=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).

% ATP.lambda_247
tff(fact_8431_ATP_Olambda__248,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_ags(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(B),set(set(product_prod(A,B))),image2(B,set(product_prod(A,B)),aTP_Lamp_agr(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_248
tff(fact_8432_ATP_Olambda__249,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_acs(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = aa(list(list(product_prod(A,B))),list(product_prod(A,B)),concat(product_prod(A,B)),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),aa(fun(product_prod(C,B),list(product_prod(A,B))),fun(list(product_prod(C,B)),list(list(product_prod(A,B)))),map(product_prod(C,B),list(product_prod(A,B))),aTP_Lamp_acr(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).

% ATP.lambda_249
tff(fact_8433_ATP_Olambda__250,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_mj(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_250
tff(fact_8434_ATP_Olambda__251,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fb(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_251
tff(fact_8435_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gf(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_252
tff(fact_8436_ATP_Olambda__253,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aeq(list(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),aa(list(A),set(A),set2(A),Uu))) ) ).

% ATP.lambda_253
tff(fact_8437_ATP_Olambda__254,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_akg(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua != last(A,Uu) ) ) ).

% ATP.lambda_254
tff(fact_8438_ATP_Olambda__255,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_agy(nat,fun(nat,set(nat)),Uu),Uua) = set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua)) ).

% ATP.lambda_255
tff(fact_8439_ATP_Olambda__256,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_qr(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_256
tff(fact_8440_ATP_Olambda__257,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_wf(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = aa(product_prod(A,B),fun(C,product_prod(product_prod(A,B),C)),product_Pair(product_prod(A,B),C),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_257
tff(fact_8441_ATP_Olambda__258,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_uk(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uu,Uua)) ) ).

% ATP.lambda_258
tff(fact_8442_ATP_Olambda__259,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_ty(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uua,Uu)) ) ).

% ATP.lambda_259
tff(fact_8443_ATP_Olambda__260,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_zv(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),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).

% ATP.lambda_260
tff(fact_8444_ATP_Olambda__261,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_agt(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),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_261
tff(fact_8445_ATP_Olambda__262,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abm(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_262
tff(fact_8446_ATP_Olambda__263,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abl(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_263
tff(fact_8447_ATP_Olambda__264,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abe(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_264
tff(fact_8448_ATP_Olambda__265,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abf(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_265
tff(fact_8449_ATP_Olambda__266,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_bv(set(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_266
tff(fact_8450_ATP_Olambda__267,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aek(A,fun(A,bool),Uu),Uua))
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_267
tff(fact_8451_ATP_Olambda__268,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aeo(A,fun(A,bool)),Uu),Uua))
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_268
tff(fact_8452_ATP_Olambda__269,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_ajd(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_269
tff(fact_8453_ATP_Olambda__270,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_fv(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_270
tff(fact_8454_ATP_Olambda__271,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_om(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_271
tff(fact_8455_ATP_Olambda__272,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: B] : aa(B,A,aTP_Lamp_jb(fun(B,bool),fun(B,A),Uu),Uua) = aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_272
tff(fact_8456_ATP_Olambda__273,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_agn(fun(B,A),fun(B,set(A)),Uu),Uua) = set_ord_atMost(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_273
tff(fact_8457_ATP_Olambda__274,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ql(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_274
tff(fact_8458_ATP_Olambda__275,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_pn(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_8459_ATP_Olambda__276,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_oo(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_8460_ATP_Olambda__277,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_us(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_277
tff(fact_8461_ATP_Olambda__278,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vi(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8462_ATP_Olambda__279,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ms(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_279
tff(fact_8463_ATP_Olambda__280,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_tn(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_280
tff(fact_8464_ATP_Olambda__281,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_281
tff(fact_8465_ATP_Olambda__282,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ro(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_282
tff(fact_8466_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_283
tff(fact_8467_ATP_Olambda__284,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_hr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_284
tff(fact_8468_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fh(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_285
tff(fact_8469_ATP_Olambda__286,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pc(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_286
tff(fact_8470_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_nq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8471_ATP_Olambda__288,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_ox(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,arcosh(real),aa(B,real,Uu,Uua)) ).

% ATP.lambda_288
tff(fact_8472_ATP_Olambda__289,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ts(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_289
tff(fact_8473_ATP_Olambda__290,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ns(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_8474_ATP_Olambda__291,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_on(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,sgn_sgn(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_291
tff(fact_8475_ATP_Olambda__292,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vj(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_292
tff(fact_8476_ATP_Olambda__293,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_to(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_293
tff(fact_8477_ATP_Olambda__294,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_fw(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_294
tff(fact_8478_ATP_Olambda__295,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_es(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_8479_ATP_Olambda__296,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [Uu: fun(A9,A9),Uua: A9] : aa(A9,A9,aTP_Lamp_nb(fun(A9,A9),fun(A9,A9),Uu),Uua) = aa(A9,A9,tanh(A9),aa(A9,A9,Uu,Uua)) ) ).

% ATP.lambda_296
tff(fact_8480_ATP_Olambda__297,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_vm(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_297
tff(fact_8481_ATP_Olambda__298,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_tp(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_298
tff(fact_8482_ATP_Olambda__299,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_ow(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_8483_ATP_Olambda__300,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ou(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_300
tff(fact_8484_ATP_Olambda__301,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_hb(fun(A,B),fun(A,B),Uu),Uua) = exp(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_8485_ATP_Olambda__302,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ov(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_302
tff(fact_8486_ATP_Olambda__303,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_abw(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_303
tff(fact_8487_ATP_Olambda__304,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_abq(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_304
tff(fact_8488_ATP_Olambda__305,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_ach(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uua)) ).

% ATP.lambda_305
tff(fact_8489_ATP_Olambda__306,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_acp(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uua)) ).

% ATP.lambda_306
tff(fact_8490_ATP_Olambda__307,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_tv(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_307
tff(fact_8491_ATP_Olambda__308,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_tw(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_308
tff(fact_8492_ATP_Olambda__309,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_ma(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_309
tff(fact_8493_ATP_Olambda__310,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_pq(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_310
tff(fact_8494_ATP_Olambda__311,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_xg(fun(B,list(A)),fun(B,set(A)),Uu),Uua) = aa(list(A),set(A),set2(A),aa(B,list(A),Uu,Uua)) ).

% ATP.lambda_311
tff(fact_8495_ATP_Olambda__312,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_nu(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_312
tff(fact_8496_ATP_Olambda__313,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_zq(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_313
tff(fact_8497_ATP_Olambda__314,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_aet(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_314
tff(fact_8498_ATP_Olambda__315,axiom,
    ! [B: $tType,Uu: fun(B,bool),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_xy(fun(B,bool),fun(B,bool),Uu),Uua))
    <=> ~ pp(aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_315
tff(fact_8499_ATP_Olambda__316,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_bu(fun(A,bool),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(A,bool,Uu,Uua)) ) ).

% ATP.lambda_316
tff(fact_8500_ATP_Olambda__317,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_ahg(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aj(nat,fun(nat,bool)),Uu)) ).

% ATP.lambda_317
tff(fact_8501_ATP_Olambda__318,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_yz(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),bool),aTP_Lamp_yy(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua))) ).

% ATP.lambda_318
tff(fact_8502_ATP_Olambda__319,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real] : aa(real,filter(A),aTP_Lamp_uj(A,fun(real,filter(A)),Uu),Uua) = principal(A,aa(fun(A,bool),set(A),collect(A),aa(real,fun(A,bool),aTP_Lamp_ui(A,fun(real,fun(A,bool)),Uu),Uua))) ) ).

% ATP.lambda_319
tff(fact_8503_ATP_Olambda__320,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alg(set(A),fun(A,bool),Uu),Uua))
        <=> ? [F7: fun(A,real)] :
              ( ( Uua = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),F7)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),F7))) )
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),F7))),Uu))
              & pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),F7)))) ) ) ) ).

% ATP.lambda_320
tff(fact_8504_ATP_Olambda__321,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alf(set(A),fun(A,bool),Uu),Uua))
        <=> ? [F7: fun(A,real)] :
              ( ( Uua = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),F7)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),F7))) )
              & pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),F7))))
              & ! [V6: A] :
                  ( ( aa(A,real,F7,V6) != zero_zero(real) )
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V6),Uu)) ) ) ) ) ).

% ATP.lambda_321
tff(fact_8505_ATP_Olambda__322,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_yt(list(A),fun(A,bool),Uu),Uua))
    <=> ? [I4: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).

% ATP.lambda_322
tff(fact_8506_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_zj(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F7),Uu) )
              & ! [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F7,X3)),X3)) ) ) ) ) ).

% ATP.lambda_323
tff(fact_8507_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_zi(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F7),Uu) )
              & ! [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F7,X3)),X3)) ) ) ) ) ).

% ATP.lambda_324
tff(fact_8508_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_zk(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F7),Uu) )
              & ! [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F7,X3)),X3)) ) ) ) ) ).

% ATP.lambda_325
tff(fact_8509_ATP_Olambda__326,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_zl(set(A),fun(set(A),bool),Uu),Uua))
    <=> ? [B10: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B10) )
          & pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Uu),pow(A,B10))) ) ) ).

% ATP.lambda_326
tff(fact_8510_ATP_Olambda__327,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_zd(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X3)) ) ) ) ).

% ATP.lambda_327
tff(fact_8511_ATP_Olambda__328,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ys(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X3)) ) ) ) ).

% ATP.lambda_328
tff(fact_8512_ATP_Olambda__329,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ze(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Uua)) ) ) ) ).

% ATP.lambda_329
tff(fact_8513_ATP_Olambda__330,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_yr(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Uua)) ) ) ) ).

% ATP.lambda_330
tff(fact_8514_ATP_Olambda__331,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_tb(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y4))
             => pp(aa(A,bool,Uu,Y4)) ) ) ) ).

% ATP.lambda_331
tff(fact_8515_ATP_Olambda__332,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ajc(set(product_prod(A,A)),fun(set(A),bool),Uu),Uua))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uua))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),Uua))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),Uu))
                | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3)),Uu)) ) ) ) ) ).

% ATP.lambda_332
tff(fact_8516_ATP_Olambda__333,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_abk(fun(A,option(B)),fun(B,bool),Uu),Uua))
    <=> ? [A7: A] : aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),Uua) ) ).

% ATP.lambda_333
tff(fact_8517_ATP_Olambda__334,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_yp(fun(B,A),fun(A,bool),Uu),Uua))
    <=> ? [X3: B] : Uua = aa(B,A,Uu,X3) ) ).

% ATP.lambda_334
tff(fact_8518_ATP_Olambda__335,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ald(set(A),fun(A,bool),Uu),Uua))
        <=> ? [T4: 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_ajz(fun(A,real),fun(A,A),R5)),T4) )
              & pp(aa(set(A),bool,finite_finite2(A),T4))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T4),Uu)) ) ) ) ).

% ATP.lambda_335
tff(fact_8519_ATP_Olambda__336,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_abg(fun(A,option(B)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> ? [A7: A,B7: B] :
          ( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
          & ( aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),B7) ) ) ) ).

% ATP.lambda_336
tff(fact_8520_ATP_Olambda__337,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
      ( pp(aa(product_prod(set(A),set(A)),bool,aTP_Lamp_ahx(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),Uu),Uua))
    <=> ? [X8: set(A),Y7: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X8),Y7) )
          & ( X8 != bot_bot(set(A)) )
          & ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Y7))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),X8))
                  & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3)),Uu)) ) ) ) ) ).

% ATP.lambda_337
tff(fact_8521_ATP_Olambda__338,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_afy(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_338
tff(fact_8522_ATP_Olambda__339,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_agh(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).

% ATP.lambda_339
tff(fact_8523_ATP_Olambda__340,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_akf(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert2(A),Uu) ).

% ATP.lambda_340
tff(fact_8524_ATP_Olambda__341,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_aic(set(A),fun(list(A),set(list(A))),Uu),Uua) = lists(A,Uu) ).

% ATP.lambda_341
tff(fact_8525_ATP_Olambda__342,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_ie(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_342
tff(fact_8526_ATP_Olambda__343,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bg(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_343
tff(fact_8527_ATP_Olambda__344,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_bh(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_344
tff(fact_8528_ATP_Olambda__345,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_be(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),Uub)) ) ).

% ATP.lambda_345
tff(fact_8529_ATP_Olambda__346,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_adm(A,fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uu),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub))) ).

% ATP.lambda_346
tff(fact_8530_ATP_Olambda__347,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_el(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_347
tff(fact_8531_ATP_Olambda__348,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_jd(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_348
tff(fact_8532_ATP_Olambda__349,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_je(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_349
tff(fact_8533_ATP_Olambda__350,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_br(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_350
tff(fact_8534_ATP_Olambda__351,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gm(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_351
tff(fact_8535_ATP_Olambda__352,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_ed(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_352
tff(fact_8536_ATP_Olambda__353,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bq(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_353
tff(fact_8537_ATP_Olambda__354,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gn(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_354
tff(fact_8538_ATP_Olambda__355,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_iv(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_355
tff(fact_8539_ATP_Olambda__356,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_iy(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).

% ATP.lambda_356
tff(fact_8540_ATP_Olambda__357,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_iu(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).

% ATP.lambda_357
tff(fact_8541_ATP_Olambda__358,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_zw(fun(A,bool),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = if(set(A),aa(A,bool,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),Uub),Uub) ).

% ATP.lambda_358
tff(fact_8542_ATP_Olambda__359,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,bool),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ek(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(nat,bool,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_359
tff(fact_8543_ATP_Olambda__360,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_ch(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_360
tff(fact_8544_ATP_Olambda__361,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_aej(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_361
tff(fact_8545_ATP_Olambda__362,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_gv(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_362
tff(fact_8546_ATP_Olambda__363,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,option(A),aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_afg(fun(B,A),fun(fun(B,bool),fun(B,option(A))),Uu),Uua),Uub) = if(option(A),aa(B,bool,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_363
tff(fact_8547_ATP_Olambda__364,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_agu(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_agt(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_364
tff(fact_8548_ATP_Olambda__365,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_aah(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_zv(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_365
tff(fact_8549_ATP_Olambda__366,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] : aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_abo(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub) = fun_upd(A,option(B),Uu,Uua,aa(B,option(B),some(B),Uub)) ).

% ATP.lambda_366
tff(fact_8550_ATP_Olambda__367,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aa(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_akc(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),Uu),Uua),Uub) = fun_upd(A,option(B),Uub,Uu,aa(B,option(B),some(B),Uua)) ).

% ATP.lambda_367
tff(fact_8551_ATP_Olambda__368,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_xf(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_368
tff(fact_8552_ATP_Olambda__369,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_hu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_369
tff(fact_8553_ATP_Olambda__370,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_dj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_370
tff(fact_8554_ATP_Olambda__371,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_gs(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_371
tff(fact_8555_ATP_Olambda__372,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),Uua) ) ).

% ATP.lambda_372
tff(fact_8556_ATP_Olambda__373,axiom,
    ! [I7: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(I7,fun(A,B)),Uua: A,Uub: I7] : aa(I7,B,aa(A,fun(I7,B),aTP_Lamp_pv(fun(I7,fun(A,B)),fun(A,fun(I7,B)),Uu),Uua),Uub) = aa(A,B,aa(I7,fun(A,B),Uu,Uub),Uua) ) ).

% ATP.lambda_373
tff(fact_8557_ATP_Olambda__374,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_gp(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_374
tff(fact_8558_ATP_Olambda__375,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_ca(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_375
tff(fact_8559_ATP_Olambda__376,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_xw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_376
tff(fact_8560_ATP_Olambda__377,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_afu(fun(B,fun(A,A)),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(B,fun(A,A),Uu,Uub),Uua) ).

% ATP.lambda_377
tff(fact_8561_ATP_Olambda__378,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_up(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_378
tff(fact_8562_ATP_Olambda__379,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_or(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_379
tff(fact_8563_ATP_Olambda__380,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,A)),Uua: B,Uub: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_afv(fun(A,fun(B,A)),fun(B,fun(A,A)),Uu),Uua),Uub) = aa(B,A,aa(A,fun(B,A),Uu,Uub),Uua) ).

% ATP.lambda_380
tff(fact_8564_ATP_Olambda__381,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aft(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_381
tff(fact_8565_ATP_Olambda__382,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_fm(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_fl(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_382
tff(fact_8566_ATP_Olambda__383,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_fk(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_fj(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_383
tff(fact_8567_ATP_Olambda__384,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_fe(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_fd(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_384
tff(fact_8568_ATP_Olambda__385,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_em(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uub),Uu),one_one(A),zero_zero(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_385
tff(fact_8569_ATP_Olambda__386,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,bool),aa(code_integer,fun(code_integer,product_prod(code_integer,bool)),aTP_Lamp_is(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),Uu),Uua),Uub) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),one_one(code_integer))) ).

% ATP.lambda_386
tff(fact_8570_ATP_Olambda__387,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_all(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_387
tff(fact_8571_ATP_Olambda__388,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_gt(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_gs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_388
tff(fact_8572_ATP_Olambda__389,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_hf(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_he(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_389
tff(fact_8573_ATP_Olambda__390,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_id(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ic(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_390
tff(fact_8574_ATP_Olambda__391,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ib(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ia(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_391
tff(fact_8575_ATP_Olambda__392,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hz(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_hy(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_392
tff(fact_8576_ATP_Olambda__393,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hx(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_hw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_393
tff(fact_8577_ATP_Olambda__394,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ht(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_hs(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_394
tff(fact_8578_ATP_Olambda__395,axiom,
    ! [A: $tType,B: $tType,I7: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I7),Uua: fun(I7,fun(A,B)),Uub: A] : aa(A,B,aa(fun(I7,fun(A,B)),fun(A,B),aTP_Lamp_pw(set(I7),fun(fun(I7,fun(A,B)),fun(A,B)),Uu),Uua),Uub) = aa(set(I7),B,aa(fun(I7,B),fun(set(I7),B),groups7121269368397514597t_prod(I7,B),aa(A,fun(I7,B),aTP_Lamp_pv(fun(I7,fun(A,B)),fun(A,fun(I7,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_395
tff(fact_8579_ATP_Olambda__396,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_uq(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_up(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_396
tff(fact_8580_ATP_Olambda__397,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_nc(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,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_397
tff(fact_8581_ATP_Olambda__398,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_nd(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,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_398
tff(fact_8582_ATP_Olambda__399,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_ez(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,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_399
tff(fact_8583_ATP_Olambda__400,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_dl(fun(nat,A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_400
tff(fact_8584_ATP_Olambda__401,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_pa(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua)),Uub) ) ).

% ATP.lambda_401
tff(fact_8585_ATP_Olambda__402,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_pe(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua)),Uub) ) ).

% ATP.lambda_402
tff(fact_8586_ATP_Olambda__403,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_bt(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,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_403
tff(fact_8587_ATP_Olambda__404,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_pf(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_404
tff(fact_8588_ATP_Olambda__405,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_nm(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_405
tff(fact_8589_ATP_Olambda__406,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_fa(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,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_406
tff(fact_8590_ATP_Olambda__407,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_aev(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_407
tff(fact_8591_ATP_Olambda__408,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_ge(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_408
tff(fact_8592_ATP_Olambda__409,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_acd(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub)))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub)),lex(A,Uu))) ) ) ) ).

% ATP.lambda_409
tff(fact_8593_ATP_Olambda__410,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_ajw(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(nat,nat,suc,Uu) )
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).

% ATP.lambda_410
tff(fact_8594_ATP_Olambda__411,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_adi(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X3: A,Y4: A,Xs6: list(A),Ys7: list(A)] :
            ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs6)) )
            & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys7)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4)),Uu)) ) ) ) ).

% ATP.lambda_411
tff(fact_8595_ATP_Olambda__412,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_aew(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_412
tff(fact_8596_ATP_Olambda__413,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(A),fun(list(A),bool),aTP_Lamp_io(nat,fun(set(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua)) ) ) ).

% ATP.lambda_413
tff(fact_8597_ATP_Olambda__414,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_in(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)) ) ) ).

% ATP.lambda_414
tff(fact_8598_ATP_Olambda__415,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_agd(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_415
tff(fact_8599_ATP_Olambda__416,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_ip(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua))) ) ) ).

% ATP.lambda_416
tff(fact_8600_ATP_Olambda__417,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_au(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua)) ) ) ).

% ATP.lambda_417
tff(fact_8601_ATP_Olambda__418,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_at(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_418
tff(fact_8602_ATP_Olambda__419,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_aeu(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_419
tff(fact_8603_ATP_Olambda__420,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,option(B))] :
      ( pp(aa(fun(A,option(B)),bool,aa(set(B),fun(fun(A,option(B)),bool),aTP_Lamp_abs(set(A),fun(set(B),fun(fun(A,option(B)),bool)),Uu),Uua),Uub))
    <=> ( ( dom(A,B,Uub) = Uu )
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),ran(A,B,Uub)),Uua)) ) ) ).

% ATP.lambda_420
tff(fact_8604_ATP_Olambda__421,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_mx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_421
tff(fact_8605_ATP_Olambda__422,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ik(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_422
tff(fact_8606_ATP_Olambda__423,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_aff(set(nat),fun(nat,fun(product_prod(A,nat),bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua)),Uu)) ) ).

% ATP.lambda_423
tff(fact_8607_ATP_Olambda__424,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),aTP_Lamp_ir(nat,fun(nat,fun(set(nat),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(nat)),bool,aa(set(nat),fun(set(set(nat)),bool),member(set(nat)),Uub),pow(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu))))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_424
tff(fact_8608_ATP_Olambda__425,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_aby(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)))
        & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uua)) ) ) ).

% ATP.lambda_425
tff(fact_8609_ATP_Olambda__426,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_aer(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))
        & pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ) ).

% ATP.lambda_426
tff(fact_8610_ATP_Olambda__427,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afr(list(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(list(A),set(A),set2(A),Uu)))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_427
tff(fact_8611_ATP_Olambda__428,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_aeh(fun(A,bool),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(list(A),set(A),set2(A),Uua)))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_428
tff(fact_8612_ATP_Olambda__429,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(A,fun(product_prod(A,B),bool),aTP_Lamp_abh(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),Uub),graph(A,B,Uu)))
        & ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).

% ATP.lambda_429
tff(fact_8613_ATP_Olambda__430,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gx(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_430
tff(fact_8614_ATP_Olambda__431,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ail(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),field2(A,Uu)))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uua))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X3)),Uu)) ) ) ) ).

% ATP.lambda_431
tff(fact_8615_ATP_Olambda__432,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ain(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),field2(A,Uu)))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uua))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Uub)),Uu)) ) ) ) ).

% ATP.lambda_432
tff(fact_8616_ATP_Olambda__433,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aim(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),field2(A,Uu)))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uua))
           => ( ( Uub != X3 )
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X3)),Uu)) ) ) ) ) ).

% ATP.lambda_433
tff(fact_8617_ATP_Olambda__434,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_ade(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))),Uua)) ) ).

% ATP.lambda_434
tff(fact_8618_ATP_Olambda__435,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ahh(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uub)) ) ) ).

% ATP.lambda_435
tff(fact_8619_ATP_Olambda__436,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_ajp(set(A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,infini527867602293511546merate(A,Uu),Uua)),Uub)) ) ) ) ).

% ATP.lambda_436
tff(fact_8620_ATP_Olambda__437,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_aho(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & ( aa(A,B,Uua,Uub) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,Uua),Uu)) ) ) ) ) ).

% ATP.lambda_437
tff(fact_8621_ATP_Olambda__438,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_ahw(set(nat),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu))
        & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ahv(set(nat),fun(nat,fun(nat,bool)),Uu),Uub)))),Uua)) ) ) ).

% ATP.lambda_438
tff(fact_8622_ATP_Olambda__439,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(nat,fun(set(A),bool),aTP_Lamp_ig(set(A),fun(nat,fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu))
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_439
tff(fact_8623_ATP_Olambda__440,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ij(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(nat,nat,suc,Uua))) ) ) ).

% ATP.lambda_440
tff(fact_8624_ATP_Olambda__441,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_gd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_441
tff(fact_8625_ATP_Olambda__442,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_mv(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_442
tff(fact_8626_ATP_Olambda__443,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_gj(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_443
tff(fact_8627_ATP_Olambda__444,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ep(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub)) ).

% ATP.lambda_444
tff(fact_8628_ATP_Olambda__445,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ye(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uua)) ) ) ).

% ATP.lambda_445
tff(fact_8629_ATP_Olambda__446,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_bx(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uua)) ) ) ).

% ATP.lambda_446
tff(fact_8630_ATP_Olambda__447,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_bm(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uua))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_447
tff(fact_8631_ATP_Olambda__448,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ac(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_448
tff(fact_8632_ATP_Olambda__449,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_bl(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_449
tff(fact_8633_ATP_Olambda__450,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ah(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_450
tff(fact_8634_ATP_Olambda__451,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ai(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_451
tff(fact_8635_ATP_Olambda__452,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ad(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_452
tff(fact_8636_ATP_Olambda__453,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ajb(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).

% ATP.lambda_453
tff(fact_8637_ATP_Olambda__454,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_af(vEBT_VEBT,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(Uu),Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uub)) ) ) ).

% ATP.lambda_454
tff(fact_8638_ATP_Olambda__455,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ae(vEBT_VEBT,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(Uu),Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_455
tff(fact_8639_ATP_Olambda__456,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_uz(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uu)) ) ) ).

% ATP.lambda_456
tff(fact_8640_ATP_Olambda__457,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ace(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu)) ) ) ).

% ATP.lambda_457
tff(fact_8641_ATP_Olambda__458,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ahv(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_458
tff(fact_8642_ATP_Olambda__459,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_jc(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uua)) ) ) ).

% ATP.lambda_459
tff(fact_8643_ATP_Olambda__460,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,real),fun(A,bool),aTP_Lamp_vo(set(A),fun(fun(A,real),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,Uua,Uub))) ) ) ) ).

% ATP.lambda_460
tff(fact_8644_ATP_Olambda__461,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_aed(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_461
tff(fact_8645_ATP_Olambda__462,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_aex(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_462
tff(fact_8646_ATP_Olambda__463,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_cg(set(B),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_463
tff(fact_8647_ATP_Olambda__464,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: set(A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ajs(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_464
tff(fact_8648_ATP_Olambda__465,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_am(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_465
tff(fact_8649_ATP_Olambda__466,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_zy(fun(A,bool),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uua))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_466
tff(fact_8650_ATP_Olambda__467,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ce(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_467
tff(fact_8651_ATP_Olambda__468,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_cd(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_468
tff(fact_8652_ATP_Olambda__469,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_agm(set(A),fun(fun(A,set(B)),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_469
tff(fact_8653_ATP_Olambda__470,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_az(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_470
tff(fact_8654_ATP_Olambda__471,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_eu(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_471
tff(fact_8655_ATP_Olambda__472,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ax(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_472
tff(fact_8656_ATP_Olambda__473,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_ev(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua))
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_473
tff(fact_8657_ATP_Olambda__474,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_if(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(B,A,Uua,Uub))) ) ) ) ).

% ATP.lambda_474
tff(fact_8658_ATP_Olambda__475,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_al(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uua)) ) ) ).

% ATP.lambda_475
tff(fact_8659_ATP_Olambda__476,axiom,
    ! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_acq(list(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_476
tff(fact_8660_ATP_Olambda__477,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ux(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uu)),Uua)) ) ).

% ATP.lambda_477
tff(fact_8661_ATP_Olambda__478,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_di(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_478
tff(fact_8662_ATP_Olambda__479,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_rf(A,fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_479
tff(fact_8663_ATP_Olambda__480,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( pp(aa(complex,bool,aa(complex,fun(complex,bool),aTP_Lamp_vr(real,fun(complex,fun(complex,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(complex,Uua,Uub)),Uu)) ) ).

% ATP.lambda_480
tff(fact_8664_ATP_Olambda__481,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),aTP_Lamp_vt(real,fun(real,fun(real,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(real,Uua,Uub)),Uu)) ) ).

% ATP.lambda_481
tff(fact_8665_ATP_Olambda__482,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_vd(real,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu)) ) ) ).

% ATP.lambda_482
tff(fact_8666_ATP_Olambda__483,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_rp(real,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu)) ) ) ).

% ATP.lambda_483
tff(fact_8667_ATP_Olambda__484,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_ui(A,fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uub,Uu)),Uua)) ) ) ).

% ATP.lambda_484
tff(fact_8668_ATP_Olambda__485,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_dm(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_485
tff(fact_8669_ATP_Olambda__486,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_lh(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uub,Uu)),Uua) ) ).

% ATP.lambda_486
tff(fact_8670_ATP_Olambda__487,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_lf(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_487
tff(fact_8671_ATP_Olambda__488,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_lg(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_488
tff(fact_8672_ATP_Olambda__489,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_le(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_489
tff(fact_8673_ATP_Olambda__490,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aji(set(product_prod(B,A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_490
tff(fact_8674_ATP_Olambda__491,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,bool))),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)),Uu)) ) ).

% ATP.lambda_491
tff(fact_8675_ATP_Olambda__492,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_wl(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_492
tff(fact_8676_ATP_Olambda__493,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aip(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ).

% ATP.lambda_493
tff(fact_8677_ATP_Olambda__494,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_adz(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_494
tff(fact_8678_ATP_Olambda__495,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa(complex,bool,aa(complex,fun(complex,bool),aTP_Lamp_av(nat,fun(complex,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu) = Uua ) ) ).

% ATP.lambda_495
tff(fact_8679_ATP_Olambda__496,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa(complex,bool,aa(nat,fun(complex,bool),aTP_Lamp_il(complex,fun(nat,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu ) ) ).

% ATP.lambda_496
tff(fact_8680_ATP_Olambda__497,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_xs(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uub))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uub),Uua)) ) ) ) ).

% ATP.lambda_497
tff(fact_8681_ATP_Olambda__498,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_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,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_498
tff(fact_8682_ATP_Olambda__499,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_nn(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_499
tff(fact_8683_ATP_Olambda__500,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_gb(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_500
tff(fact_8684_ATP_Olambda__501,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dp(fun(nat,nat),fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),Uub)) ).

% ATP.lambda_501
tff(fact_8685_ATP_Olambda__502,axiom,
    ! [Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(nat,Aa),Uua: Aa,Uub: nat] : aa(nat,Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_tz(fun(nat,Aa),fun(Aa,fun(nat,Aa)),Uu),Uua),Uub) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uu,Uub)),aa(nat,Aa,aa(Aa,fun(nat,Aa),power_power(Aa),Uua),Uub)) ) ).

% ATP.lambda_502
tff(fact_8686_ATP_Olambda__503,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_dg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_503
tff(fact_8687_ATP_Olambda__504,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_504
tff(fact_8688_ATP_Olambda__505,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_fx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_505
tff(fact_8689_ATP_Olambda__506,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_gi(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_506
tff(fact_8690_ATP_Olambda__507,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_dh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_507
tff(fact_8691_ATP_Olambda__508,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_508
tff(fact_8692_ATP_Olambda__509,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ar(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,Uu,Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_509
tff(fact_8693_ATP_Olambda__510,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,A)),Uub: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_agq(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uua,Uub)),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_510
tff(fact_8694_ATP_Olambda__511,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,A)),Uub: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_agp(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uua,Uub)),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_511
tff(fact_8695_ATP_Olambda__512,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_afn(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_512
tff(fact_8696_ATP_Olambda__513,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_sa(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_513
tff(fact_8697_ATP_Olambda__514,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_se(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_514
tff(fact_8698_ATP_Olambda__515,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_wx(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ) ).

% ATP.lambda_515
tff(fact_8699_ATP_Olambda__516,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_sn(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uua,Uub)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_516
tff(fact_8700_ATP_Olambda__517,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_pt(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = divide_divide(A,aa(C,A,Uu,Uub),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_517
tff(fact_8701_ATP_Olambda__518,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_pl(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = divide_divide(A,aa(C,A,Uu,Uub),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_518
tff(fact_8702_ATP_Olambda__519,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_rl(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = divide_divide(A,aa(C,A,Uu,Uub),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_519
tff(fact_8703_ATP_Olambda__520,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_op(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_520
tff(fact_8704_ATP_Olambda__521,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_ur(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_521
tff(fact_8705_ATP_Olambda__522,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_vg(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_522
tff(fact_8706_ATP_Olambda__523,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_mr(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_523
tff(fact_8707_ATP_Olambda__524,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_tm(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_524
tff(fact_8708_ATP_Olambda__525,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_ss(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_525
tff(fact_8709_ATP_Olambda__526,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_vk(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ) ).

% ATP.lambda_526
tff(fact_8710_ATP_Olambda__527,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aij(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_527
tff(fact_8711_ATP_Olambda__528,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_hn(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_528
tff(fact_8712_ATP_Olambda__529,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_hm(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_529
tff(fact_8713_ATP_Olambda__530,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D] : aa(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_oj(fun(D,A),fun(fun(D,A),fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),aa(D,A,Uua,Uub)) ) ).

% ATP.lambda_530
tff(fact_8714_ATP_Olambda__531,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rk(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8715_ATP_Olambda__532,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_jv(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8716_ATP_Olambda__533,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_ri(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_533
tff(fact_8717_ATP_Olambda__534,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rh(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_534
tff(fact_8718_ATP_Olambda__535,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_et(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_535
tff(fact_8719_ATP_Olambda__536,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_of(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_536
tff(fact_8720_ATP_Olambda__537,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_en(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_537
tff(fact_8721_ATP_Olambda__538,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oh(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uua,Uub)),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_538
tff(fact_8722_ATP_Olambda__539,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_cy(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_539
tff(fact_8723_ATP_Olambda__540,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_ci(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_540
tff(fact_8724_ATP_Olambda__541,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yu(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_541
tff(fact_8725_ATP_Olambda__542,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uw(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_542
tff(fact_8726_ATP_Olambda__543,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_oi(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_543
tff(fact_8727_ATP_Olambda__544,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ew(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_544
tff(fact_8728_ATP_Olambda__545,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_na(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_545
tff(fact_8729_ATP_Olambda__546,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [Uu: fun(C,real),Uua: fun(C,real),Uub: C] : aa(C,real,aa(fun(C,real),fun(C,real),aTP_Lamp_vp(fun(C,real),fun(fun(C,real),fun(C,real)),Uu),Uua),Uub) = powr(real,aa(C,real,Uu,Uub),aa(C,real,Uua,Uub)) ) ).

% ATP.lambda_546
tff(fact_8730_ATP_Olambda__547,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_pz(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_547
tff(fact_8731_ATP_Olambda__548,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_sx(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_548
tff(fact_8732_ATP_Olambda__549,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_vq(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_549
tff(fact_8733_ATP_Olambda__550,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_tt(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_550
tff(fact_8734_ATP_Olambda__551,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_pb(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_551
tff(fact_8735_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,product_prod(B,B),aa(A,fun(A,product_prod(B,B)),aTP_Lamp_agv(fun(A,B),fun(A,fun(A,product_prod(B,B))),Uu),Uua),Uub) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uu,Uua)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_552
tff(fact_8736_ATP_Olambda__553,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_wj(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).

% ATP.lambda_553
tff(fact_8737_ATP_Olambda__554,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_vf(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_554
tff(fact_8738_ATP_Olambda__555,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_tl(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_555
tff(fact_8739_ATP_Olambda__556,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_oc(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_556
tff(fact_8740_ATP_Olambda__557,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_wb(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ).

% ATP.lambda_557
tff(fact_8741_ATP_Olambda__558,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_yf(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_558
tff(fact_8742_ATP_Olambda__559,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_akl(fun(nat,set(A)),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(nat,set(A),Uu,Uua)),aa(nat,set(A),Uu,Uub))) ) ).

% ATP.lambda_559
tff(fact_8743_ATP_Olambda__560,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_akk(fun(A,set(B)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),aa(A,set(B),Uu,Uua)),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_560
tff(fact_8744_ATP_Olambda__561,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_aaj(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_561
tff(fact_8745_ATP_Olambda__562,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_akz(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(A,bool,Uu,Uub))
            | pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_562
tff(fact_8746_ATP_Olambda__563,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ab(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        | pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_563
tff(fact_8747_ATP_Olambda__564,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_564
tff(fact_8748_ATP_Olambda__565,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aeg(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_565
tff(fact_8749_ATP_Olambda__566,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ael(fun(A,B),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_566
tff(fact_8750_ATP_Olambda__567,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_afs(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).

% ATP.lambda_567
tff(fact_8751_ATP_Olambda__568,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_ja(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uua,Uub))) ) ).

% ATP.lambda_568
tff(fact_8752_ATP_Olambda__569,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_xq(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua)))) ) ) ).

% ATP.lambda_569
tff(fact_8753_ATP_Olambda__570,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_tg(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & ! [Y4: A] :
            ( pp(aa(A,bool,Uua,Y4))
           => pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Y4)) ) ) ) ).

% ATP.lambda_570
tff(fact_8754_ATP_Olambda__571,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,set(A),aa(A,fun(B,set(A)),aTP_Lamp_ago(fun(B,A),fun(A,fun(B,set(A))),Uu),Uua),Uub) = set_or3652927894154168847AtMost(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_571
tff(fact_8755_ATP_Olambda__572,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_os(set(B),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_572
tff(fact_8756_ATP_Olambda__573,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( pp(aa(nat,bool,aa(set(A),fun(nat,bool),aTP_Lamp_sh(fun(nat,set(A)),fun(set(A),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_573
tff(fact_8757_ATP_Olambda__574,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_as(fun(nat,nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua)) ) ).

% ATP.lambda_574
tff(fact_8758_ATP_Olambda__575,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ls(fun(B,set(A)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_575
tff(fact_8759_ATP_Olambda__576,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_sp(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_576
tff(fact_8760_ATP_Olambda__577,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_sl(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_577
tff(fact_8761_ATP_Olambda__578,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ry(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_578
tff(fact_8762_ATP_Olambda__579,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_rt(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_579
tff(fact_8763_ATP_Olambda__580,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_fr(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uu,Uub),Uua) ) ).

% ATP.lambda_580
tff(fact_8764_ATP_Olambda__581,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_jh(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_581
tff(fact_8765_ATP_Olambda__582,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_oq(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),Uua) ) ).

% 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_qf(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uua,Uub),Uu) ) ).

% ATP.lambda_583
tff(fact_8767_ATP_Olambda__584,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_sc(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_584
tff(fact_8768_ATP_Olambda__585,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_sq(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_585
tff(fact_8769_ATP_Olambda__586,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ru(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_586
tff(fact_8770_ATP_Olambda__587,axiom,
    ! [D: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D,A),Uua: A,Uub: D] : aa(D,A,aa(A,fun(D,A),aTP_Lamp_ok(fun(D,A),fun(A,fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_587
tff(fact_8771_ATP_Olambda__588,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_uo(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_588
tff(fact_8772_ATP_Olambda__589,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_ef(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_589
tff(fact_8773_ATP_Olambda__590,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_qe(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_590
tff(fact_8774_ATP_Olambda__591,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_nz(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_591
tff(fact_8775_ATP_Olambda__592,axiom,
    ! [E: $tType,F: $tType,Uu: fun(E,set(F)),Uua: set(F),Uub: E] : aa(E,set(F),aa(set(F),fun(E,set(F)),aTP_Lamp_me(fun(E,set(F)),fun(set(F),fun(E,set(F))),Uu),Uua),Uub) = aa(set(F),set(F),aa(set(F),fun(set(F),set(F)),minus_minus(set(F)),aa(E,set(F),Uu,Uub)),Uua) ).

% ATP.lambda_592
tff(fact_8776_ATP_Olambda__593,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_og(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_593
tff(fact_8777_ATP_Olambda__594,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_mt(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_594
tff(fact_8778_ATP_Olambda__595,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_rm(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_595
tff(fact_8779_ATP_Olambda__596,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_oz(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_596
tff(fact_8780_ATP_Olambda__597,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_rs(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_597
tff(fact_8781_ATP_Olambda__598,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_oa(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_598
tff(fact_8782_ATP_Olambda__599,axiom,
    ! [C: $tType,D: $tType,Uu: fun(C,set(D)),Uua: set(D),Uub: C] : aa(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_xz(fun(C,set(D)),fun(set(D),fun(C,set(D))),Uu),Uua),Uub) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(C,set(D),Uu,Uub)),Uua) ).

% ATP.lambda_599
tff(fact_8783_ATP_Olambda__600,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_la(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_600
tff(fact_8784_ATP_Olambda__601,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_lv(fun(A,set(B)),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),Uua) ).

% ATP.lambda_601
tff(fact_8785_ATP_Olambda__602,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_uy(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_602
tff(fact_8786_ATP_Olambda__603,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_mz(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_603
tff(fact_8787_ATP_Olambda__604,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rj(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_604
tff(fact_8788_ATP_Olambda__605,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_wa(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_605
tff(fact_8789_ATP_Olambda__606,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,A),Uua: int,Uub: C] : aa(C,A,aa(int,fun(C,A),aTP_Lamp_xj(fun(C,A),fun(int,fun(C,A)),Uu),Uua),Uub) = power_int(A,aa(C,A,Uu,Uub),Uua) ) ).

% ATP.lambda_606
tff(fact_8790_ATP_Olambda__607,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(B,A),Uua: int,Uub: B] : aa(B,A,aa(int,fun(B,A),aTP_Lamp_xm(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_607
tff(fact_8791_ATP_Olambda__608,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_xo(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,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_xl(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_609
tff(fact_8793_ATP_Olambda__610,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_xp(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_610
tff(fact_8794_ATP_Olambda__611,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_xn(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_611
tff(fact_8795_ATP_Olambda__612,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_ahc(fun(A,B),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,Uu,Uub)),Uua)) ) ).

% ATP.lambda_612
tff(fact_8796_ATP_Olambda__613,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_xa(set(A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uua,Uub)),Uu)) ) ) ).

% ATP.lambda_613
tff(fact_8797_ATP_Olambda__614,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ww(set(A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uua,Uub)),Uu)) ) ).

% ATP.lambda_614
tff(fact_8798_ATP_Olambda__615,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ajx(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_615
tff(fact_8799_ATP_Olambda__616,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aif(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_616
tff(fact_8800_ATP_Olambda__617,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ahe(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_617
tff(fact_8801_ATP_Olambda__618,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aio(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uua )
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu)) ) ) ).

% ATP.lambda_618
tff(fact_8802_ATP_Olambda__619,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ahj(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uua )
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ) ).

% ATP.lambda_619
tff(fact_8803_ATP_Olambda__620,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_bw(A,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_620
tff(fact_8804_ATP_Olambda__621,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gw(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub))) ) ).

% ATP.lambda_621
tff(fact_8805_ATP_Olambda__622,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_gc(real,fun(fun(nat,A),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_622
tff(fact_8806_ATP_Olambda__623,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_iz(fun(B,bool),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uub))),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_623
tff(fact_8807_ATP_Olambda__624,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_ta(fun(A,B),fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua)) ) ) ).

% ATP.lambda_624
tff(fact_8808_ATP_Olambda__625,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_xr(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_625
tff(fact_8809_ATP_Olambda__626,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_zx(fun(A,A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( ? [X3: A] :
                ( ( Uub = aa(A,A,Uu,X3) )
                & pp(aa(A,bool,Uua,X3)) )
            | ? [M9: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M9) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M9)
                & ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),M9))
                   => pp(aa(A,bool,Uua,X3)) ) ) ) ) ) ).

% ATP.lambda_626
tff(fact_8810_ATP_Olambda__627,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: fun(A,real)] :
          ( pp(aa(fun(A,real),bool,aa(A,fun(fun(A,real),bool),aTP_Lamp_alh(set(A),fun(A,fun(fun(A,real),bool)),Uu),Uua),Uub))
        <=> ( ! [V6: A] :
                ( ( aa(A,real,Uub,V6) != zero_zero(real) )
               => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V6),Uu)) )
            & pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),Uub))))
            & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_ajz(fun(A,real),fun(A,A),Uub)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aka(fun(A,real),fun(A,bool),Uub))) = Uua ) ) ) ) ).

% ATP.lambda_627
tff(fact_8811_ATP_Olambda__628,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),bool)),Uua: list(A),Uub: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_adb(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub))
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
            | ? [X3: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4)) )
            | ? [X3: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X3))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uu,Xs3),Ys4)) ) ) ) ) ).

% ATP.lambda_628
tff(fact_8812_ATP_Olambda__629,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_yy(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).

% ATP.lambda_629
tff(fact_8813_ATP_Olambda__630,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ahp(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uua))
        & pp(aa(set(A),bool,finite_finite2(A),Uub))
        & ( Uub != bot_bot(set(A)) )
        & ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uua))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),Uub))
                & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3)),Uu)) ) ) ) ) ).

% ATP.lambda_630
tff(fact_8814_ATP_Olambda__631,axiom,
    ! [A: $tType,Uu: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),Uua: fun(bool,fun(bool,A)),Uub: vEBT_VEBT] : aa(vEBT_VEBT,product_prod(vEBT_VEBT,A),aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_aky(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),Uu),Uua),Uub) = aa(A,product_prod(vEBT_VEBT,A),aa(vEBT_VEBT,fun(A,product_prod(vEBT_VEBT,A)),product_Pair(vEBT_VEBT,A),Uub),vEBT_rec_VEBT(A,Uu,Uua,Uub)) ).

% ATP.lambda_631
tff(fact_8815_ATP_Olambda__632,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_hh(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_632
tff(fact_8816_ATP_Olambda__633,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_xi(A,fun(int,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uua)),power_int(A,Uu,aa(int,int,aa(int,fun(int,int),minus_minus(int),Uua),one_one(int))))) ) ).

% ATP.lambda_633
tff(fact_8817_ATP_Olambda__634,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cp(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] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cq(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(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_635
tff(fact_8819_ATP_Olambda__636,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aaf(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A)))))) ) ).

% ATP.lambda_636
tff(fact_8820_ATP_Olambda__637,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_qq(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = divide_divide(real,Uua,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_637
tff(fact_8821_ATP_Olambda__638,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_bn(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_638
tff(fact_8822_ATP_Olambda__639,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: A,Uua: B,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(B,fun(C,product_prod(A,product_prod(B,C))),aa(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_acm(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uu),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).

% ATP.lambda_639
tff(fact_8823_ATP_Olambda__640,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: B,Uua: A,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(A,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_acn(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uua),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uub)) ).

% ATP.lambda_640
tff(fact_8824_ATP_Olambda__641,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_acz(A,fun(list(A),fun(nat,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),aa(list(A),list(A),aa(nat,fun(list(A),list(A)),take(A),Uub),Uua)) ).

% ATP.lambda_641
tff(fact_8825_ATP_Olambda__642,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] : aa(A,list(A),aa(list(B),fun(A,list(A)),aTP_Lamp_afb(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),map_filter(B,A,Uu,Uua)) ).

% ATP.lambda_642
tff(fact_8826_ATP_Olambda__643,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),aTP_Lamp_si(real,fun(real,fun(real,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Uub),set_or5935395276787703475ssThan(real,Uu,Uua))) ) ).

% ATP.lambda_643
tff(fact_8827_ATP_Olambda__644,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ajt(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),aa(set(A),set(B),image2(A,B,Uu),Uua))) ) ) ).

% ATP.lambda_644
tff(fact_8828_ATP_Olambda__645,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_lz(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),Uua),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_645
tff(fact_8829_ATP_Olambda__646,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ly(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),Uua),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_646
tff(fact_8830_ATP_Olambda__647,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_sk(A,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_647
tff(fact_8831_ATP_Olambda__648,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_so(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_648
tff(fact_8832_ATP_Olambda__649,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_sm(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_649
tff(fact_8833_ATP_Olambda__650,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_sd(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_650
tff(fact_8834_ATP_Olambda__651,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_sb(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_651
tff(fact_8835_ATP_Olambda__652,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_afw(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_652
tff(fact_8836_ATP_Olambda__653,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_sf(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_653
tff(fact_8837_ATP_Olambda__654,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_uh(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_654
tff(fact_8838_ATP_Olambda__655,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_eg(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] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_un(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,
    ! [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_ee(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_657
tff(fact_8841_ATP_Olambda__658,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_qd(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_658
tff(fact_8842_ATP_Olambda__659,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_ny(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_659
tff(fact_8843_ATP_Olambda__660,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D,A),Uua: A,Uub: D] : aa(D,A,aa(A,fun(D,A),aTP_Lamp_ol(fun(D,A),fun(A,fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(D,A,Uu,Uub)) ) ).

% ATP.lambda_660
tff(fact_8844_ATP_Olambda__661,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ug(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_661
tff(fact_8845_ATP_Olambda__662,axiom,
    ! [G2: $tType,H3: $tType,Uu: set(G2),Uua: fun(H3,set(G2)),Uub: H3] : aa(H3,set(G2),aa(fun(H3,set(G2)),fun(H3,set(G2)),aTP_Lamp_lx(set(G2),fun(fun(H3,set(G2)),fun(H3,set(G2))),Uu),Uua),Uub) = aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),Uu),aa(H3,set(G2),Uua,Uub)) ).

% ATP.lambda_662
tff(fact_8846_ATP_Olambda__663,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(B,nat),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_qu(fun(B,nat),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(B,nat,Uu,Uub)) ) ).

% ATP.lambda_663
tff(fact_8847_ATP_Olambda__664,axiom,
    ! [E: $tType,F: $tType,Uu: set(E),Uua: fun(F,set(E)),Uub: F] : aa(F,set(E),aa(fun(F,set(E)),fun(F,set(E)),aTP_Lamp_ya(set(E),fun(fun(F,set(E)),fun(F,set(E))),Uu),Uua),Uub) = aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),Uu),aa(F,set(E),Uua,Uub)) ).

% ATP.lambda_664
tff(fact_8848_ATP_Olambda__665,axiom,
    ! [C: $tType,D: $tType,Uu: set(C),Uua: fun(D,set(C)),Uub: D] : aa(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_lu(set(C),fun(fun(D,set(C)),fun(D,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),Uu),aa(D,set(C),Uua,Uub)) ).

% ATP.lambda_665
tff(fact_8849_ATP_Olambda__666,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_lb(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_666
tff(fact_8850_ATP_Olambda__667,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ji(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_667
tff(fact_8851_ATP_Olambda__668,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_we(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_668
tff(fact_8852_ATP_Olambda__669,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_aci(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu,Uub)) ).

% ATP.lambda_669
tff(fact_8853_ATP_Olambda__670,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_ajk(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = image(B,A,Uu,aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_670
tff(fact_8854_ATP_Olambda__671,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_lq(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_671
tff(fact_8855_ATP_Olambda__672,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_wy(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_672
tff(fact_8856_ATP_Olambda__673,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_va(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub)) ) ).

% ATP.lambda_673
tff(fact_8857_ATP_Olambda__674,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: bool] :
      ( pp(aa(bool,bool,aa(A,fun(bool,bool),aTP_Lamp_ahu(fun(A,bool),fun(A,fun(bool,bool)),Uu),Uua),Uub))
    <=> ( pp(Uub)
        | pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_674
tff(fact_8858_ATP_Olambda__675,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: bool] :
      ( pp(aa(bool,bool,aa(A,fun(bool,bool),aTP_Lamp_zr(fun(A,bool),fun(A,fun(bool,bool)),Uu),Uua),Uub))
    <=> ( pp(Uub)
        & pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_675
tff(fact_8859_ATP_Olambda__676,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_afo(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_676
tff(fact_8860_ATP_Olambda__677,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_xd(fun(A,B),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_677
tff(fact_8861_ATP_Olambda__678,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hi(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_678
tff(fact_8862_ATP_Olambda__679,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_aem(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(list(A),set(A),set2(A),nths(A,Uu,Uua)))) ) ).

% ATP.lambda_679
tff(fact_8863_ATP_Olambda__680,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(C,A),Uua: real,Uub: C] :
          ( pp(aa(C,bool,aa(real,fun(C,bool),aTP_Lamp_sv(fun(C,A),fun(real,fun(C,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(A,aa(C,A,Uu,Uub)))) ) ) ).

% ATP.lambda_680
tff(fact_8864_ATP_Olambda__681,axiom,
    ! [A: $tType,Uu: list(A),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_afa(list(A),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uub),Uu) ).

% ATP.lambda_681
tff(fact_8865_ATP_Olambda__682,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_agi(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_682
tff(fact_8866_ATP_Olambda__683,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_agl(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uu),Uua) ).

% ATP.lambda_683
tff(fact_8867_ATP_Olambda__684,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qk(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_684
tff(fact_8868_ATP_Olambda__685,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_oy(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_685
tff(fact_8869_ATP_Olambda__686,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ga(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_686
tff(fact_8870_ATP_Olambda__687,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,bool),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_sj(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua))) ) ) ).

% ATP.lambda_687
tff(fact_8871_ATP_Olambda__688,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_pk(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_688
tff(fact_8872_ATP_Olambda__689,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),bool),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ahr(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(A,B),bool,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub))) ) ).

% ATP.lambda_689
tff(fact_8873_ATP_Olambda__690,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_bf(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_690
tff(fact_8874_ATP_Olambda__691,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),bool),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_vn(fun(product_prod(A,A),bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(product_prod(A,A),bool,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua))) ) ) ).

% ATP.lambda_691
tff(fact_8875_ATP_Olambda__692,axiom,
    ! [D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D) )
     => ! [Uu: A,Uua: fun(A,D),Uub: A] : aa(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_pd(A,fun(fun(A,D),fun(A,D)),Uu),Uua),Uub) = aa(A,D,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_692
tff(fact_8876_ATP_Olambda__693,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tu(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_693
tff(fact_8877_ATP_Olambda__694,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ej(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu)) ) ).

% ATP.lambda_694
tff(fact_8878_ATP_Olambda__695,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_acc(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_695
tff(fact_8879_ATP_Olambda__696,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(B,bool)),Uua: fun(A,C),Uub: A] : aa(A,fun(B,bool),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_aiw(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Uu),Uua),Uub) = aa(C,fun(B,bool),Uu,aa(A,C,Uua,Uub)) ).

% ATP.lambda_696
tff(fact_8880_ATP_Olambda__697,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_mh(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_697
tff(fact_8881_ATP_Olambda__698,axiom,
    ! [C: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(B,C),Uua: fun(nat,B),Uub: nat] : aa(nat,C,aa(fun(nat,B),fun(nat,C),aTP_Lamp_ut(fun(B,C),fun(fun(nat,B),fun(nat,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(nat,B,Uua,Uub)) ) ).

% ATP.lambda_698
tff(fact_8882_ATP_Olambda__699,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_ws(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ) ).

% ATP.lambda_699
tff(fact_8883_ATP_Olambda__700,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,B),Uub: A] : aa(A,C,aa(fun(A,B),fun(A,C),aTP_Lamp_yd(fun(B,C),fun(fun(A,B),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(A,B,Uua,Uub)) ).

% ATP.lambda_700
tff(fact_8884_ATP_Olambda__701,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_kq(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_701
tff(fact_8885_ATP_Olambda__702,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_aii(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_702
tff(fact_8886_ATP_Olambda__703,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_wp(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_703
tff(fact_8887_ATP_Olambda__704,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,bool),Uua: fun(nat,A),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_um(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(nat,A,Uua,Uub))) ) ) ).

% ATP.lambda_704
tff(fact_8888_ATP_Olambda__705,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_ph(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_705
tff(fact_8889_ATP_Olambda__706,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_qg(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_706
tff(fact_8890_ATP_Olambda__707,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_vh(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_707
tff(fact_8891_ATP_Olambda__708,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_ajg(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_708
tff(fact_8892_ATP_Olambda__709,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_wm(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,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,bool),aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_aix(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),Uu),Uua),Uub) = aa(A,fun(B,bool),Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_710
tff(fact_8894_ATP_Olambda__711,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aav(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,
    ! [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_aca(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_712
tff(fact_8896_ATP_Olambda__713,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_acb(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_713
tff(fact_8897_ATP_Olambda__714,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ahm(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_714
tff(fact_8898_ATP_Olambda__715,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ahl(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_715
tff(fact_8899_ATP_Olambda__716,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_mi(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_716
tff(fact_8900_ATP_Olambda__717,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_pg(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_717
tff(fact_8901_ATP_Olambda__718,axiom,
    ! [D: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(D) )
     => ! [Uu: fun(A,C),Uua: fun(C,D),Uub: A] : aa(A,D,aa(fun(C,D),fun(A,D),aTP_Lamp_tx(fun(A,C),fun(fun(C,D),fun(A,D)),Uu),Uua),Uub) = aa(C,D,Uua,aa(A,C,Uu,Uub)) ) ).

% ATP.lambda_718
tff(fact_8902_ATP_Olambda__719,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_rn(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_719
tff(fact_8903_ATP_Olambda__720,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_aha(fun(A,B),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,Uua,aa(A,B,Uu,Uub))) ) ).

% ATP.lambda_720
tff(fact_8904_ATP_Olambda__721,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_li(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_721
tff(fact_8905_ATP_Olambda__722,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_wc(fun(B,bool),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_722
tff(fact_8906_ATP_Olambda__723,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_ais(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_air(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ).

% ATP.lambda_723
tff(fact_8907_ATP_Olambda__724,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),bool),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_vx(fun(product_prod(A,A),bool),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_vw(fun(product_prod(A,A),bool),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ) ).

% ATP.lambda_724
tff(fact_8908_ATP_Olambda__725,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A] : aa(A,Aa,aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_ub(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),Uu),Uua),Uub) = suminf(Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_ua(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub)) ) ).

% ATP.lambda_725
tff(fact_8909_ATP_Olambda__726,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_nx(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_nw(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_726
tff(fact_8910_ATP_Olambda__727,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aln(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,real,real_V7696804695334737415tation(A,real_V4986007116245087402_basis(A,Uu),Uua),Uub) != zero_zero(real) ) ) ) ).

% ATP.lambda_727
tff(fact_8911_ATP_Olambda__728,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_alk(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,real,real_V7696804695334737415tation(A,Uu,Uua),Uub) != zero_zero(real) ) ) ) ).

% ATP.lambda_728
tff(fact_8912_ATP_Olambda__729,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_ud(A,fun(set(A),fun(A,filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),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_729
tff(fact_8913_ATP_Olambda__730,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_uc(A,fun(set(A),fun(A,filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,Uub)),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_730
tff(fact_8914_ATP_Olambda__731,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_tj(A,fun(set(A),fun(set(A),filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uub),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_731
tff(fact_8915_ATP_Olambda__732,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_ft(fun(nat,A),fun(A,fun(nat,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub))) ) ).

% ATP.lambda_732
tff(fact_8916_ATP_Olambda__733,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aar(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) != Uua ) ) ).

% ATP.lambda_733
tff(fact_8917_ATP_Olambda__734,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_aas(B,fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uua,Uub) != Uu ) ) ).

% ATP.lambda_734
tff(fact_8918_ATP_Olambda__735,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_vc(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub))) ) ).

% ATP.lambda_735
tff(fact_8919_ATP_Olambda__736,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_vb(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub))) ) ).

% ATP.lambda_736
tff(fact_8920_ATP_Olambda__737,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_jj(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_737
tff(fact_8921_ATP_Olambda__738,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: list(B)] : aa(list(B),fun(list(A),list(A)),aa(B,fun(list(B),fun(list(A),list(A))),aTP_Lamp_akd(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),Uu),Uua),Uub) = aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_738
tff(fact_8922_ATP_Olambda__739,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_ii(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ih(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub))) ).

% ATP.lambda_739
tff(fact_8923_ATP_Olambda__740,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_abi(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [I4: nat] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu),I4)),aa(nat,B,nth(B,Uua),I4)) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),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_740
tff(fact_8924_ATP_Olambda__741,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_agx(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [A7: A] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),aa(A,B,Uua,A7)) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),Uu)) ) ) ).

% ATP.lambda_741
tff(fact_8925_ATP_Olambda__742,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_abx(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [I4: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu)))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),I4),Uua)) ) ) ).

% ATP.lambda_742
tff(fact_8926_ATP_Olambda__743,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aad(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A7) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),Uu)) ) ) ) ).

% ATP.lambda_743
tff(fact_8927_ATP_Olambda__744,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aab(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A7) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),Uu)) ) ) ) ).

% ATP.lambda_744
tff(fact_8928_ATP_Olambda__745,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_yi(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: B] :
          ( ( Uub = aa(B,A,Uu,X3) )
          & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),Uua)) ) ) ).

% ATP.lambda_745
tff(fact_8929_ATP_Olambda__746,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_yh(fun(B,A),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: B] :
          ( ( Uub = aa(B,A,Uu,X3) )
          & pp(aa(B,bool,Uua,X3)) ) ) ).

% ATP.lambda_746
tff(fact_8930_ATP_Olambda__747,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,B),Uub: B] :
      ( pp(aa(B,bool,aa(fun(A,B),fun(B,bool),aTP_Lamp_yc(fun(A,bool),fun(fun(A,B),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( ( Uub = aa(A,B,Uua,X3) )
          & pp(aa(A,bool,Uu,X3)) ) ) ).

% ATP.lambda_747
tff(fact_8931_ATP_Olambda__748,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_yq(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uu))
         => pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),X3)) ) ) ).

% ATP.lambda_748
tff(fact_8932_ATP_Olambda__749,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_ahq(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uu))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),X3)) ) ) ).

% ATP.lambda_749
tff(fact_8933_ATP_Olambda__750,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ajj(set(product_prod(A,B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uua))
          & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Uub)),Uu)) ) ) ).

% ATP.lambda_750
tff(fact_8934_ATP_Olambda__751,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aht(fun(A,B),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uua))
          & ( aa(A,B,Uu,X3) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_751
tff(fact_8935_ATP_Olambda__752,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ahy(fun(A,option(B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uua))
          & ( aa(A,option(B),Uu,X3) = aa(B,option(B),some(B),Uub) ) ) ) ).

% ATP.lambda_752
tff(fact_8936_ATP_Olambda__753,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_sy(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uub),N3))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,Uub,N3)))),aa(nat,real,Uua,Uub))) ) ) ) ).

% ATP.lambda_753
tff(fact_8937_ATP_Olambda__754,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ahs(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uua))
          & ( Uub = aa(A,B,Uu,X3) ) ) ) ).

% ATP.lambda_754
tff(fact_8938_ATP_Olambda__755,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_td(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [A7: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uub),A7))
             => ! [B7: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A7),B7))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or3652927894154168847AtMost(nat,A7,B7)))),aa(nat,real,Uua,A7))) ) ) ) ) ).

% ATP.lambda_755
tff(fact_8939_ATP_Olambda__756,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_xv(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [Y4: A] :
          ( pp(aa(A,bool,Uu,Y4))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),Y4)) ) ) ).

% ATP.lambda_756
tff(fact_8940_ATP_Olambda__757,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_zf(fun(A,A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ? [N3: 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)),N3),Uu),Uua) ) ).

% ATP.lambda_757
tff(fact_8941_ATP_Olambda__758,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_adh(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [A7: A,V6: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),V6)) )
          | ? [U6: list(A),Aa3: A,B7: A,Va4: list(A),W4: list(A)] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa3),B7)),Uu))
              & ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa3),Va4)) )
              & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B7),W4)) ) ) ) ) ).

% ATP.lambda_758
tff(fact_8942_ATP_Olambda__759,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aae(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A7: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A7),B7) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),Uu))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B7),Uua)) ) ) ) ).

% ATP.lambda_759
tff(fact_8943_ATP_Olambda__760,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aac(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A7: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A7),B7) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),Uu))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B7),Uua)) ) ) ) ).

% ATP.lambda_760
tff(fact_8944_ATP_Olambda__761,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(list(A)),fun(list(A),bool),aTP_Lamp_add(set(A),fun(set(list(A)),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [X3: A,Xs3: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uu))
          & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs3),Uua)) ) ) ).

% ATP.lambda_761
tff(fact_8945_ATP_Olambda__762,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_adg(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [Us2: list(A),Z2: A,Z9: A,Vs2: list(A)] :
          ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Vs2)) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Z9)),Uu))
          & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z9),Vs2)) ) ) ) ).

% ATP.lambda_762
tff(fact_8946_ATP_Olambda__763,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_fl(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uuc)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),binomial(Uub,Uuc)))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_763
tff(fact_8947_ATP_Olambda__764,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_fd(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uuc))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),binomial(Uub,Uuc)))),semiring_char_0_fact(real,Uub)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_764
tff(fact_8948_ATP_Olambda__765,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_fj(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),binomial(Uub,Uuc)))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_765
tff(fact_8949_ATP_Olambda__766,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_aag(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),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),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_766
tff(fact_8950_ATP_Olambda__767,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_zo(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),aa(set(A),set(B),image2(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_767
tff(fact_8951_ATP_Olambda__768,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_ajn(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = if(set(B),aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),Uuc),Uuc) ).

% ATP.lambda_768
tff(fact_8952_ATP_Olambda__769,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_ds(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_769
tff(fact_8953_ATP_Olambda__770,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_hj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_770
tff(fact_8954_ATP_Olambda__771,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_ul(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_771
tff(fact_8955_ATP_Olambda__772,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_hk(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_772
tff(fact_8956_ATP_Olambda__773,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_dt(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_773
tff(fact_8957_ATP_Olambda__774,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_yv(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_774
tff(fact_8958_ATP_Olambda__775,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_eo(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uuc),Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_775
tff(fact_8959_ATP_Olambda__776,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_hg(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_776
tff(fact_8960_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_cf(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_777
tff(fact_8961_ATP_Olambda__778,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_im(B,fun(fun(B,A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),Uub) ) ).

% ATP.lambda_778
tff(fact_8962_ATP_Olambda__779,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: B,Uub: set(A),Uuc: A] : aa(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_ahb(B,fun(B,fun(set(A),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uub),Uu,Uua) ).

% ATP.lambda_779
tff(fact_8963_ATP_Olambda__780,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A),Uuc: list(A)] : aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_agb(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = if(product_prod(list(A),list(A)),aa(A,bool,Uu,Uua),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uuc),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uuc))) ).

% ATP.lambda_780
tff(fact_8964_ATP_Olambda__781,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_jg(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_781
tff(fact_8965_ATP_Olambda__782,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_jf(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_782
tff(fact_8966_ATP_Olambda__783,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_xx(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_783
tff(fact_8967_ATP_Olambda__784,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_aao(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_aan(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_784
tff(fact_8968_ATP_Olambda__785,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_aam(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_aal(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_785
tff(fact_8969_ATP_Olambda__786,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_jp(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_786
tff(fact_8970_ATP_Olambda__787,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,fun(list(B),fun(C,C))),Uua: fun(A,B),Uub: A,Uuc: list(A)] : aa(list(A),fun(C,C),aa(A,fun(list(A),fun(C,C)),aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_ake(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Uu),Uua),Uub),Uuc) = aa(list(B),fun(C,C),aa(B,fun(list(B),fun(C,C)),Uu,aa(A,B,Uua,Uub)),aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),Uua),Uuc)) ).

% ATP.lambda_787
tff(fact_8971_ATP_Olambda__788,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_afm(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ).

% ATP.lambda_788
tff(fact_8972_ATP_Olambda__789,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(D,C),Uuc: D] : aa(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_acg(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),Uu),Uua),Uub),Uuc) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(D,C,Uub,Uuc)) ).

% ATP.lambda_789
tff(fact_8973_ATP_Olambda__790,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_zh(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_790
tff(fact_8974_ATP_Olambda__791,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,C),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_aiv(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),aa(B,C,Uua,Uuc))) ) ).

% ATP.lambda_791
tff(fact_8975_ATP_Olambda__792,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,B),Uub: A,Uuc: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_aiy(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(C,B,Uua,Uuc))) ) ).

% ATP.lambda_792
tff(fact_8976_ATP_Olambda__793,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,A)),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,A,aa(A,fun(C,A),aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_acf(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(B,A,aa(A,fun(B,A),Uu,Uub),aa(C,B,Uua,Uuc)) ).

% ATP.lambda_793
tff(fact_8977_ATP_Olambda__794,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_cv(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_cu(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_794
tff(fact_8978_ATP_Olambda__795,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_du(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),zero_zero(nat)),aa(A,A,uminus_uminus(A),Uub),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),zero_zero(A)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_795
tff(fact_8979_ATP_Olambda__796,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_alm(set(A),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,real_V7696804695334737415tation(A,real_V4986007116245087402_basis(A,Uu),Uub),Uuc)),if(B,aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu),aa(A,B,Uua,Uuc),zero_zero(B))) ) ).

% ATP.lambda_796
tff(fact_8980_ATP_Olambda__797,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,fun(A,bool)),Uub: set(A),Uuc: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_alp(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,pred_chain(A,Uu,Uua),Uuc))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uub),Uuc)) ) ) ).

% ATP.lambda_797
tff(fact_8981_ATP_Olambda__798,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: C] : aa(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_gq(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(C,fun(B,A),aTP_Lamp_gp(fun(B,fun(C,A)),fun(C,fun(B,A)),Uua),Uuc)),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_cb(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_798
tff(fact_8982_ATP_Olambda__799,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: C] : aa(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_cc(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(C,fun(B,A),aTP_Lamp_ca(fun(B,fun(C,A)),fun(C,fun(B,A)),Uua),Uuc)),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_cb(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_799
tff(fact_8983_ATP_Olambda__800,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_dr(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_dq(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_800
tff(fact_8984_ATP_Olambda__801,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_do(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_dn(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_801
tff(fact_8985_ATP_Olambda__802,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_ni(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,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_802
tff(fact_8986_ATP_Olambda__803,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_ng(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,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_803
tff(fact_8987_ATP_Olambda__804,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_nf(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,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uub),Uua)),Uuc)) ).

% ATP.lambda_804
tff(fact_8988_ATP_Olambda__805,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_ne(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,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uua),Uub)),Uuc)) ).

% ATP.lambda_805
tff(fact_8989_ATP_Olambda__806,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: set(A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_akb(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),product_Sigma(A,A,Uua,aTP_Lamp_agj(set(A),fun(A,set(A)),Uua))))
        & pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uuc)) ) ) ).

% ATP.lambda_806
tff(fact_8990_ATP_Olambda__807,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_jt(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).

% ATP.lambda_807
tff(fact_8991_ATP_Olambda__808,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_cu(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Uuc))) ) ).

% ATP.lambda_808
tff(fact_8992_ATP_Olambda__809,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_qw(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu,Uua)),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_809
tff(fact_8993_ATP_Olambda__810,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_mo(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uua)) ) ) ) ).

% ATP.lambda_810
tff(fact_8994_ATP_Olambda__811,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(B,fun(real,fun(A,bool)),aTP_Lamp_sz(fun(A,B),fun(B,fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub)) ) ) ).

% ATP.lambda_811
tff(fact_8995_ATP_Olambda__812,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: real,Uuc: B] :
          ( pp(aa(B,bool,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_sr(fun(B,A),fun(A,fun(real,fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(B,A,Uu,Uuc),Uua)),Uub)) ) ) ).

% ATP.lambda_812
tff(fact_8996_ATP_Olambda__813,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_adk(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X3: A,Y4: A,Xs6: list(A),Ys7: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs6)) )
            & ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys7)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4)),Uu)) ) ) ) ).

% ATP.lambda_813
tff(fact_8997_ATP_Olambda__814,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_kr(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),aa(set(B),set(A),image2(B,A,Uu),Uua)))
        & pp(aa(A,bool,Uub,Uuc)) ) ) ).

% ATP.lambda_814
tff(fact_8998_ATP_Olambda__815,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(B,fun(C,bool)),Uub: B,Uuc: C] :
      ( pp(aa(C,bool,aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_by(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Uuc),Uu))
        & pp(aa(C,bool,aa(B,fun(C,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_815
tff(fact_8999_ATP_Olambda__816,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ih(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_816
tff(fact_9000_ATP_Olambda__817,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,fun(C,bool)),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_cb(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
        & pp(aa(C,bool,aa(B,fun(C,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_817
tff(fact_9001_ATP_Olambda__818,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,bool)),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_an(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_818
tff(fact_9002_ATP_Olambda__819,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: fun(A,real),Uuc: A] :
          ( pp(aa(A,bool,aa(fun(A,real),fun(A,bool),aa(fun(A,real),fun(fun(A,real),fun(A,bool)),aTP_Lamp_vl(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,Uua,Uuc)),aa(A,real,Uub,Uuc))) ) ) ) ).

% ATP.lambda_819
tff(fact_9003_ATP_Olambda__820,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_kp(set(A),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_820
tff(fact_9004_ATP_Olambda__821,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,C),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_kw(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
        & ( aa(B,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_821
tff(fact_9005_ATP_Olambda__822,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_lj(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_822
tff(fact_9006_ATP_Olambda__823,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_ahf(fun(A,B),fun(set(A),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uua))
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_823
tff(fact_9007_ATP_Olambda__824,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( pp(aa(B,bool,aa(set(B),fun(B,bool),aa(fun(B,A),fun(set(B),fun(B,bool)),aTP_Lamp_wv(set(A),fun(fun(B,A),fun(set(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uub))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uua,Uuc)),Uu)) ) ) ).

% ATP.lambda_824
tff(fact_9008_ATP_Olambda__825,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_jz(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_825
tff(fact_9009_ATP_Olambda__826,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_hs(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).

% ATP.lambda_826
tff(fact_9010_ATP_Olambda__827,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_jx(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_827
tff(fact_9011_ATP_Olambda__828,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_kb(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ).

% ATP.lambda_828
tff(fact_9012_ATP_Olambda__829,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_kd(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).

% ATP.lambda_829
tff(fact_9013_ATP_Olambda__830,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: list(A),Uub: B,Uuc: list(B)] : aa(list(B),list(product_prod(A,B)),aa(B,fun(list(B),list(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_ads(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uub)),zip(A,B,Uua,Uuc)) ).

% ATP.lambda_830
tff(fact_9014_ATP_Olambda__831,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: list(B),Uub: A,Uuc: list(A)] : aa(list(A),list(product_prod(A,B)),aa(A,fun(list(A),list(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_adt(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uu)),zip(A,B,Uuc,Uua)) ).

% ATP.lambda_831
tff(fact_9015_ATP_Olambda__832,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_gg(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_832
tff(fact_9016_ATP_Olambda__833,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: B] :
      ( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_ks(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uua))
        & pp(aa(A,bool,Uub,aa(B,A,Uu,Uuc))) ) ) ).

% ATP.lambda_833
tff(fact_9017_ATP_Olambda__834,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),aTP_Lamp_lr(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)),aa(C,set(product_prod(A,B)),Uu,Uua))) ) ).

% ATP.lambda_834
tff(fact_9018_ATP_Olambda__835,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ay(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != one_one(A) ) ) ) ) ).

% ATP.lambda_835
tff(fact_9019_ATP_Olambda__836,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ba(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_836
tff(fact_9020_ATP_Olambda__837,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_qb(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,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))) ) ).

% ATP.lambda_837
tff(fact_9021_ATP_Olambda__838,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_hl(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_838
tff(fact_9022_ATP_Olambda__839,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_nw(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_839
tff(fact_9023_ATP_Olambda__840,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A,Uuc: nat] : aa(nat,Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_ua(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub),Uuc) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uua,Uuc)),aa(nat,Aa,aa(Aa,fun(nat,Aa),power_power(Aa),aa(A,Aa,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_840
tff(fact_9024_ATP_Olambda__841,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_nv(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),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_841
tff(fact_9025_ATP_Olambda__842,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_dc(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_842
tff(fact_9026_ATP_Olambda__843,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: set(A),Uuc: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aa(A,fun(set(A),fun(B,bool)),aTP_Lamp_su(fun(B,A),fun(A,fun(set(A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,Uu,Uuc)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uub),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A)))))) ) ) ).

% ATP.lambda_843
tff(fact_9027_ATP_Olambda__844,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_aiu(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uuc))
        & ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).

% ATP.lambda_844
tff(fact_9028_ATP_Olambda__845,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_afl(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_845
tff(fact_9029_ATP_Olambda__846,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_dq(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uuc)),aa(nat,nat,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_846
tff(fact_9030_ATP_Olambda__847,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_dn(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_847
tff(fact_9031_ATP_Olambda__848,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_vy(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_848
tff(fact_9032_ATP_Olambda__849,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_acj(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).

% ATP.lambda_849
tff(fact_9033_ATP_Olambda__850,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,D),Uub: A,Uuc: B] : aa(B,product_prod(C,D),aa(A,fun(B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_ajy(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),Uu),Uua),Uub),Uuc) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,Uu,Uub)),aa(B,D,Uua,Uuc)) ).

% ATP.lambda_850
tff(fact_9034_ATP_Olambda__851,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_afq(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_851
tff(fact_9035_ATP_Olambda__852,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(C),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: B] : aa(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_go(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uua,Uuc)),aa(fun(C,bool),set(C),collect(C),aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_by(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_852
tff(fact_9036_ATP_Olambda__853,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(C),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: B] : aa(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_bz(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uua,Uuc)),aa(fun(C,bool),set(C),collect(C),aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_by(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_853
tff(fact_9037_ATP_Olambda__854,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_ps(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_854
tff(fact_9038_ATP_Olambda__855,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_nr(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% ATP.lambda_855
tff(fact_9039_ATP_Olambda__856,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_nt(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ) ).

% ATP.lambda_856
tff(fact_9040_ATP_Olambda__857,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_qz(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,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu,Uub))),aa(A,B,Uua,Uuc))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_857
tff(fact_9041_ATP_Olambda__858,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_akm(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(B,A,Uua,Uub) != aa(B,A,Uua,Uuc) )
       => pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ) ).

% ATP.lambda_858
tff(fact_9042_ATP_Olambda__859,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_zg(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),Uub),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Uu,Uuc)),Uua)) ) ) ).

% ATP.lambda_859
tff(fact_9043_ATP_Olambda__860,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_st(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uuc))),Uub))) ) ) ).

% ATP.lambda_860
tff(fact_9044_ATP_Olambda__861,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_rd(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_rc(A,A)))))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_rc(A,A))))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_rc(A,A)))))) ) ).

% ATP.lambda_861
tff(fact_9045_ATP_Olambda__862,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_rb(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))) ) ).

% ATP.lambda_862
tff(fact_9046_ATP_Olambda__863,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_ra(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))) ) ).

% ATP.lambda_863
tff(fact_9047_ATP_Olambda__864,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_lk(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,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_lj(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_864
tff(fact_9048_ATP_Olambda__865,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(list(A),fun(list(A),bool)),Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_ada(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [Y4: A,Ys4: list(A)] :
            ( ( Uub = nil(A) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
        | ? [X3: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,X3),Y4)) )
        | ? [X3: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,X3),Y4))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,Y4),X3))
            & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uua,Xs3),Ys4)) ) ) ) ).

% ATP.lambda_865
tff(fact_9049_ATP_Olambda__866,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: nat] : aa(nat,list(A),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_acy(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),aa(A,list(A),aa(nat,fun(A,list(A)),aa(list(A),fun(nat,fun(A,list(A))),list_update(A),Uua),Uuc),Uub)) ).

% ATP.lambda_866
tff(fact_9050_ATP_Olambda__867,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_agz(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_867
tff(fact_9051_ATP_Olambda__868,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_ky(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),Uua),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_kw(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_868
tff(fact_9052_ATP_Olambda__869,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_lc(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),Uub),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_kw(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_869
tff(fact_9053_ATP_Olambda__870,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_kx(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),Uua),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_kw(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_870
tff(fact_9054_ATP_Olambda__871,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_kz(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),Uub),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_kw(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_871
tff(fact_9055_ATP_Olambda__872,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_rr(fun(nat,A),fun(nat,fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),set_ord_atMost(nat,Uua))))) ) ) ).

% ATP.lambda_872
tff(fact_9056_ATP_Olambda__873,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_adn(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),ord_min(A),Uu),min_list(A,Uua)) ) ).

% ATP.lambda_873
tff(fact_9057_ATP_Olambda__874,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,fun(list(A),B)),Uub: A,Uuc: list(A)] : aa(list(A),C,aa(A,fun(list(A),C),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_adl(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),Uu),Uua),Uub),Uuc) = aa(B,C,Uu,aa(list(A),B,aa(A,fun(list(A),B),Uua,Uub),Uuc)) ).

% ATP.lambda_874
tff(fact_9058_ATP_Olambda__875,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,B),Uub: D,Uuc: C] : aa(C,A,aa(D,fun(C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_ack(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uua,Uub)),Uuc)) ).

% ATP.lambda_875
tff(fact_9059_ATP_Olambda__876,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(product_prod(B,B),bool),Uuc: A] :
          ( pp(aa(A,bool,aa(fun(product_prod(B,B),bool),fun(A,bool),aa(B,fun(fun(product_prod(B,B),bool),fun(A,bool)),aTP_Lamp_vv(fun(A,B),fun(B,fun(fun(product_prod(B,B),bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(product_prod(B,B),bool,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uu,Uuc)),Uua))) ) ) ).

% ATP.lambda_876
tff(fact_9060_ATP_Olambda__877,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,A,aa(B,fun(D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_acl(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),aa(D,C,Uua,Uuc))) ).

% ATP.lambda_877
tff(fact_9061_ATP_Olambda__878,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_uf(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ue(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_878
tff(fact_9062_ATP_Olambda__879,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_ti(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_th(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_879
tff(fact_9063_ATP_Olambda__880,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_vz(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_vy(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).

% ATP.lambda_880
tff(fact_9064_ATP_Olambda__881,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_hy(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_881
tff(fact_9065_ATP_Olambda__882,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_hw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_882
tff(fact_9066_ATP_Olambda__883,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_afx(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),takeWhile(A),aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),Uu)),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).

% ATP.lambda_883
tff(fact_9067_ATP_Olambda__884,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: C,Uub: fun(C,A),Uuc: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(C,fun(fun(C,A),fun(C,A)),aTP_Lamp_po(fun(C,A),fun(C,fun(fun(C,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua))),aa(C,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua)))) ) ).

% ATP.lambda_884
tff(fact_9068_ATP_Olambda__885,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_aeb(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).

% ATP.lambda_885
tff(fact_9069_ATP_Olambda__886,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ic(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_886
tff(fact_9070_ATP_Olambda__887,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ia(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_887
tff(fact_9071_ATP_Olambda__888,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_ajo(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)),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_888
tff(fact_9072_ATP_Olambda__889,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aa(B,fun(fun(A,B),bool),aa(set(B),fun(B,fun(fun(A,B),bool)),aTP_Lamp_tc(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ! [X3: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uu))
           => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,Uuc,X3)),Uua)) )
          & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Uu))
           => ( aa(A,B,Uuc,X3) = Uub ) ) ) ) ).

% ATP.lambda_889
tff(fact_9073_ATP_Olambda__890,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_aak(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [Y4: C] :
          ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y4)),Uu))
          & pp(aa(set(product_prod(C,B)),bool,aa(product_prod(C,B),fun(set(product_prod(C,B)),bool),member(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y4),Uuc)),Uua)) ) ) ).

% ATP.lambda_890
tff(fact_9074_ATP_Olambda__891,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_aap(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A7: C] :
          ( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A7)),aa(C,B,Uub,A7)) )
          & pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),A7),Uu)) ) ) ).

% ATP.lambda_891
tff(fact_9075_ATP_Olambda__892,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( pp(aa(C,bool,aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_yb(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [X3: A,Y4: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X3),Y4) )
          & pp(aa(A,bool,Uu,X3))
          & pp(aa(B,bool,Uua,Y4)) ) ) ).

% ATP.lambda_892
tff(fact_9076_ATP_Olambda__893,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_aan(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),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),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_893
tff(fact_9077_ATP_Olambda__894,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_aal(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(A,C)),aa(B,bool,aa(B,fun(B,bool),fequal(B),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),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_894
tff(fact_9078_ATP_Olambda__895,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_pi(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_895
tff(fact_9079_ATP_Olambda__896,axiom,
    ! [A: $tType,B: $tType,I7: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I7),Uua: fun(I7,fun(A,B)),Uub: fun(I7,fun(A,B)),Uuc: A,Uud: A] : aa(A,B,aa(A,fun(A,B),aa(fun(I7,fun(A,B)),fun(A,fun(A,B)),aa(fun(I7,fun(A,B)),fun(fun(I7,fun(A,B)),fun(A,fun(A,B))),aTP_Lamp_py(set(I7),fun(fun(I7,fun(A,B)),fun(fun(I7,fun(A,B)),fun(A,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(I7),B,aa(fun(I7,B),fun(set(I7),B),groups7311177749621191930dd_sum(I7,B),aa(A,fun(I7,B),aa(A,fun(A,fun(I7,B)),aa(fun(I7,fun(A,B)),fun(A,fun(A,fun(I7,B))),aa(fun(I7,fun(A,B)),fun(fun(I7,fun(A,B)),fun(A,fun(A,fun(I7,B)))),aTP_Lamp_px(set(I7),fun(fun(I7,fun(A,B)),fun(fun(I7,fun(A,B)),fun(A,fun(A,fun(I7,B))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_896
tff(fact_9080_ATP_Olambda__897,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_dw(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_dv(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_897
tff(fact_9081_ATP_Olambda__898,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_nh(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_ng(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uud),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)))))) ).

% ATP.lambda_898
tff(fact_9082_ATP_Olambda__899,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_dd(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_dc(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud)) ) ).

% ATP.lambda_899
tff(fact_9083_ATP_Olambda__900,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_zc(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_trancl(A,Uub)))
          | ( Uuc = Uu ) )
        & ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_trancl(A,Uub)))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_900
tff(fact_9084_ATP_Olambda__901,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_air(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),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))))))),field2(A,Uu)))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id(A))))
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id(A)))) )
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & ( Uua = Uuc )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id(A)))) ) ) ) ) ).

% ATP.lambda_901
tff(fact_9085_ATP_Olambda__902,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_zb(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_rtrancl(A,Uub)))
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_rtrancl(A,Uub))) ) ) ).

% ATP.lambda_902
tff(fact_9086_ATP_Olambda__903,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_dv(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uud)),one_one(nat)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_903
tff(fact_9087_ATP_Olambda__904,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_xu(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uuc),Uud))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uud),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu)))
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))
           => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I4)),X_13))
            <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Uub),I4)) ) )
        & ( ( Uuc = Uud )
         => ! [X3: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua)))
             => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_13)) ) )
        & ( ( Uuc != Uud )
         => ( vEBT_V5917875025757280293ildren(divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua,Uud)
            & ! [X3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu)))
               => ( vEBT_V5917875025757280293ildren(divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua,X3)
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),X3))
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_904
tff(fact_9088_ATP_Olambda__905,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(product_prod(B,B),bool),Uuc: A,Uud: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(product_prod(B,B),bool),fun(A,fun(A,bool)),aa(fun(A,B),fun(fun(product_prod(B,B),bool),fun(A,fun(A,bool))),aTP_Lamp_xb(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),bool),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uud),Uu))
             => pp(aa(product_prod(B,B),bool,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uuc)),aa(A,B,Uua,Uud)))) ) ) ) ) ).

% ATP.lambda_905
tff(fact_9089_ATP_Olambda__906,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),bool),Uua: A,Uub: A,Uuc: A,Uud: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_vw(fun(product_prod(A,A),bool),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ( ( Uub = Uuc )
           => pp(aa(product_prod(A,A),bool,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud))) ) ) ) ).

% ATP.lambda_906
tff(fact_9090_ATP_Olambda__907,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,A),Uua: C,Uub: fun(C,A),Uuc: int,Uud: C] : aa(C,A,aa(int,fun(C,A),aa(fun(C,A),fun(int,fun(C,A)),aa(C,fun(fun(C,A),fun(int,fun(C,A))),aTP_Lamp_xk(fun(C,A),fun(C,fun(fun(C,A),fun(int,fun(C,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uub,Uud)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uuc)),power_int(A,aa(C,A,Uu,Uua),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uuc),one_one(int))))) ) ).

% ATP.lambda_907
tff(fact_9091_ATP_Olambda__908,axiom,
    ! [A: $tType,B: $tType,I7: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I7),Uua: fun(I7,fun(A,B)),Uub: fun(I7,fun(A,B)),Uuc: A,Uud: A,Uue: I7] : aa(I7,B,aa(A,fun(I7,B),aa(A,fun(A,fun(I7,B)),aa(fun(I7,fun(A,B)),fun(A,fun(A,fun(I7,B))),aa(fun(I7,fun(A,B)),fun(fun(I7,fun(A,B)),fun(A,fun(A,fun(I7,B)))),aTP_Lamp_px(set(I7),fun(fun(I7,fun(A,B)),fun(fun(I7,fun(A,B)),fun(A,fun(A,fun(I7,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,aa(I7,fun(A,B),Uub,Uue),Uud)),aa(set(I7),B,aa(fun(I7,B),fun(set(I7),B),groups7121269368397514597t_prod(I7,B),aa(A,fun(I7,B),aTP_Lamp_pv(fun(I7,fun(A,B)),fun(A,fun(I7,B)),Uua),Uuc)),aa(set(I7),set(I7),aa(set(I7),fun(set(I7),set(I7)),minus_minus(set(I7)),Uu),aa(set(I7),set(I7),aa(I7,fun(set(I7),set(I7)),insert2(I7),Uue),bot_bot(set(I7)))))) ) ).

% ATP.lambda_908
tff(fact_9092_ATP_Olambda__909,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_pm(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uua,Uue)),aa(C,A,Uuc,Uub))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uu,Uub)),aa(C,A,Uud,Uue))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uuc,Uub)),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_909
tff(fact_9093_ATP_Olambda__910,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_th(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),Uu))
        | ( ( Uub = Uud )
          & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),Uua)) ) ) ) ).

% ATP.lambda_910
tff(fact_9094_ATP_Olambda__911,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_qa(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_911
tff(fact_9095_ATP_Olambda__912,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_pu(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(C,A,Uu,Uub))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(C,A,Uuc,Uub))),aa(C,A,Uud,Uue))),aa(A,A,inverse_inverse(A),aa(C,A,Uuc,Uub))))),divide_divide(A,aa(C,A,Uua,Uue),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_912
tff(fact_9096_ATP_Olambda__913,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ue(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( ( Uub = Uud )
        & pp(aa(A,bool,Uu,Uud))
        & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),aa(A,set(product_prod(B,B)),Uua,Uud))) ) ) ).

% ATP.lambda_913
tff(fact_9097_ATP_Olambda__914,axiom,
    ! [B: $tType,A: $tType,Uu: bool,Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_mm(bool,fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(Uu) ) ).

% ATP.lambda_914
tff(fact_9098_ATP_Olambda__915,axiom,
    ! [A: $tType,Uu: bool,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_md(bool,fun(A,bool),Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_915
tff(fact_9099_ATP_Olambda__916,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_agk(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_916
tff(fact_9100_ATP_Olambda__917,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_aki(set(B),fun(A,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_917
tff(fact_9101_ATP_Olambda__918,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_agg(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_918
tff(fact_9102_ATP_Olambda__919,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_agw(set(A),fun(A,set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_919
tff(fact_9103_ATP_Olambda__920,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_age(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_920
tff(fact_9104_ATP_Olambda__921,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_lo(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_921
tff(fact_9105_ATP_Olambda__922,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_agj(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_922
tff(fact_9106_ATP_Olambda__923,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,fun(B,bool),aTP_Lamp_wd(fun(B,bool),fun(A,fun(B,bool)),Uu),Uua) = Uu ).

% ATP.lambda_923
tff(fact_9107_ATP_Olambda__924,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_aai(fun(B,B),fun(A,fun(B,B)),Uu),Uua) = Uu ).

% ATP.lambda_924
tff(fact_9108_ATP_Olambda__925,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( zero(Aa)
        & topological_t2_space(Aa)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: Aa,Uua: A] : aa(A,Aa,aTP_Lamp_ob(Aa,fun(A,Aa),Uu),Uua) = Uu ) ).

% ATP.lambda_925
tff(fact_9109_ATP_Olambda__926,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_od(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_926
tff(fact_9110_ATP_Olambda__927,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_aig(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_927
tff(fact_9111_ATP_Olambda__928,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_aes(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_928
tff(fact_9112_ATP_Olambda__929,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ku(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_929
tff(fact_9113_ATP_Olambda__930,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_kn(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_930
tff(fact_9114_ATP_Olambda__931,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ll(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_931
tff(fact_9115_ATP_Olambda__932,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_km(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_932
tff(fact_9116_ATP_Olambda__933,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fs(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_933
tff(fact_9117_ATP_Olambda__934,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mq(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_934
tff(fact_9118_ATP_Olambda__935,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_kl(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_935
tff(fact_9119_ATP_Olambda__936,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ot(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_936
tff(fact_9120_ATP_Olambda__937,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_aih(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_937
tff(fact_9121_ATP_Olambda__938,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_uv(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_938
tff(fact_9122_ATP_Olambda__939,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_aga(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_939
tff(fact_9123_ATP_Olambda__940,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_adv(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_940
tff(fact_9124_ATP_Olambda__941,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_kv(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_941
tff(fact_9125_ATP_Olambda__942,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_aez(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_942
tff(fact_9126_ATP_Olambda__943,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_adq(A,fun(list(A),bool)),Uu),Uua))
    <=> $false ) ).

% ATP.lambda_943
tff(fact_9127_ATP_Olambda__944,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_adr(A,fun(list(A),bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_944
tff(fact_9128_ATP_Olambda__945,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_afp(A,fun(A,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_945
tff(fact_9129_ATP_Olambda__946,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_cr(complex,complex),Uu) = Uu ).

% ATP.lambda_946
tff(fact_9130_ATP_Olambda__947,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_co(nat,nat),Uu) = Uu ).

% ATP.lambda_947
tff(fact_9131_ATP_Olambda__948,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_aie(int,int),Uu) = Uu ).

% ATP.lambda_948
tff(fact_9132_ATP_Olambda__949,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [Uu: C] : aa(C,C,aTP_Lamp_tr(C,C),Uu) = Uu ) ).

% ATP.lambda_949
tff(fact_9133_ATP_Olambda__950,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rc(A,A),Uu) = Uu ) ).

% ATP.lambda_950
tff(fact_9134_ATP_Olambda__951,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rg(A,A),Uu) = Uu ) ).

% ATP.lambda_951
tff(fact_9135_ATP_Olambda__952,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_tq(A,A),Uu) = Uu ) ).

% ATP.lambda_952
tff(fact_9136_ATP_Olambda__953,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aen(A,A),Uu) = Uu ) ).

% ATP.lambda_953
tff(fact_9137_ATP_Olambda__954,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_kg(A,A),Uu) = Uu ) ).

% ATP.lambda_954
tff(fact_9138_ATP_Olambda__955,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ki(A,A),Uu) = Uu ).

% ATP.lambda_955
tff(fact_9139_ATP_Olambda__956,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_lm(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_956
tff(fact_9140_ATP_Olambda__957,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_lt(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_957
tff(fact_9141_ATP_Olambda__958,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_kk(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_958
tff(fact_9142_ATP_Olambda__959,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_kj(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_959
tff(fact_9143_ATP_Olambda__960,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_ln(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_960
tff(fact_9144_ATP_Olambda__961,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_agf(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_961
tff(fact_9145_ATP_Olambda__962,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ec(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_962
tff(fact_9146_ATP_Olambda__963,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ff(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_963
tff(fact_9147_ATP_Olambda__964,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_bo(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_964
tff(fact_9148_ATP_Olambda__965,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_aei(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_965
tff(fact_9149_ATP_Olambda__966,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_oe(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_966
tff(fact_9150_ATP_Olambda__967,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A] : aa(A,real,aTP_Lamp_ali(A,real),Uu) = zero_zero(real) ) ).

% ATP.lambda_967
tff(fact_9151_ATP_Olambda__968,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_alq(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_968
tff(fact_9152_ATP_Olambda__969,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ap(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_969
tff(fact_9153_ATP_Olambda__970,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ex(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_970
tff(fact_9154_ATP_Olambda__971,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_abj(B,option(A)),Uu) = none(A) ).

% ATP.lambda_971
tff(fact_9155_ATP_Olambda__972,axiom,
    ! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_abv(A,option(C)),Uu) = none(C) ).

% ATP.lambda_972
tff(fact_9156_ATP_Olambda__973,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_aat(A,option(B)),Uu) = none(B) ).

% ATP.lambda_973
tff(fact_9157_ATP_Olambda__974,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_agc(A,B),Uu) = undefined(B) ).

% ATP.lambda_974
tff(fact_9158_ATP_Olambda__975,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_abc(nat,bool),Uu))
    <=> $false ) ).

% ATP.lambda_975
tff(fact_9159_ATP_Olambda__976,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_ao(A,bool),Uu))
    <=> $false ) ).

% ATP.lambda_976
tff(fact_9160_ATP_Olambda__977,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_abd(nat,bool),Uu))
    <=> $true ) ).

% ATP.lambda_977
tff(fact_9161_ATP_Olambda__978,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_mf(A,bool),Uu))
    <=> $true ) ).

% ATP.lambda_978
tff(fact_9162_ATP_Olambda__979,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(nat,nat),aTP_Lamp_mb(B,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_979
tff(fact_9163_ATP_Olambda__980,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_mg(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_980

% Type constructors (785)
tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple592849572758109894attice(A20)
     => counta4013691401010221786attice(fun(A9,A20)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => condit1219197933456340205attice(fun(A9,A20)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( counta3822494911875563373attice(A20)
     => counta3822494911875563373attice(fun(A9,A20)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple592849572758109894attice(A20)
     => comple592849572758109894attice(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A9: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounde4967611905675639751up_bot(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A9: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounde4346867609351753570nf_top(fun(A9,A20)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple6319245703460814977attice(fun(A9,A20)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A9: $tType,A20: $tType] :
      ( boolea8198339166811842893lgebra(A20)
     => boolea8198339166811842893lgebra(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A9: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounded_lattice_bot(fun(A9,A20)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple9053668089753744459l_ccpo(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A9: $tType,A20: $tType] :
      ( semilattice_sup(A20)
     => semilattice_sup(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A9: $tType,A20: $tType] :
      ( semilattice_inf(A20)
     => semilattice_inf(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( distrib_lattice(A20)
     => distrib_lattice(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounded_lattice(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A9: $tType,A20: $tType] :
      ( order_top(A20)
     => order_top(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A9: $tType,A20: $tType] :
      ( order_bot(A20)
     => order_bot(fun(A9,A20)) ) ).

tff(tcon_fun___Countable_Ocountable,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( finite_finite(A9)
        & countable(A20) )
     => countable(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A9: $tType,A20: $tType] :
      ( preorder(A20)
     => preorder(fun(A9,A20)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( finite_finite(A9)
        & finite_finite(A20) )
     => finite_finite(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( lattice(A20)
     => lattice(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A9: $tType,A20: $tType] :
      ( order(A20)
     => order(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A9: $tType,A20: $tType] :
      ( top(A20)
     => top(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ord(A20)
     => ord(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A9: $tType,A20: $tType] :
      ( bot(A20)
     => bot(fun(A9,A20)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A9: $tType,A20: $tType] :
      ( uminus(A20)
     => uminus(fun(A9,A20)) ) ).

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_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_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___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_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_11,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_12,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_13,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_14,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_15,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_16,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_17,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_18,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_19,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_20,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_21,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_22,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_23,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_24,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_25,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_26,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_27,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_28,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_29,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_30,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_31,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_32,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_33,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_34,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_35,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_36,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_37,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_38,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_39,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_40,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_41,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_42,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_43,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_44,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_45,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_46,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_47,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_48,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_49,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_50,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_51,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_52,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_53,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_54,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_55,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_56,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_57,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_58,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_59,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_60,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_61,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_62,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_63,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_64,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_65,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_66,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_67,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_68,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_69,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_70,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_71,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_72,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_73,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Countable_Ocountable_74,axiom,
    countable(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_75,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_76,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_77,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_78,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_79,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_80,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_81,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_82,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_83,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_84,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_85,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_86,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_87,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_88,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_89,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_90,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Power_Opower_91,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_92,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_93,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_94,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_95,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_96,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_97,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_98,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_99,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_100,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_101,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_102,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_103,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_104,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_105,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_106,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_107,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_108,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_109,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_110,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_111,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_112,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_113,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_114,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_115,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_116,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_117,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_118,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_119,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_120,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_121,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_122,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_123,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_124,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_125,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_126,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_127,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_128,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_129,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_130,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_131,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_132,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_133,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_134,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_135,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_136,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_137,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_138,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_139,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_140,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_141,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_142,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_143,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_144,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_145,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_146,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_147,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_148,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_149,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_150,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_151,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_152,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_153,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Countable_Ocountable_154,axiom,
    countable(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_155,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_156,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_157,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_158,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_159,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_160,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_161,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_162,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_163,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_164,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_165,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_166,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_167,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_168,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_169,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_170,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_171,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_172,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_173,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_174,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_175,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_176,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_177,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_178,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_179,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_180,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_181,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_182,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_183,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_184,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_185,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_186,axiom,
    ! [A9: $tType] : counta4013691401010221786attice(set(A9)) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_187,axiom,
    ! [A9: $tType] : condit1219197933456340205attice(set(A9)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_188,axiom,
    ! [A9: $tType] : counta3822494911875563373attice(set(A9)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_189,axiom,
    ! [A9: $tType] : comple592849572758109894attice(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_190,axiom,
    ! [A9: $tType] : bounde4967611905675639751up_bot(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_191,axiom,
    ! [A9: $tType] : bounde4346867609351753570nf_top(set(A9)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_192,axiom,
    ! [A9: $tType] : comple6319245703460814977attice(set(A9)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_193,axiom,
    ! [A9: $tType] : boolea8198339166811842893lgebra(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_194,axiom,
    ! [A9: $tType] : bounded_lattice_bot(set(A9)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_195,axiom,
    ! [A9: $tType] : comple9053668089753744459l_ccpo(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_196,axiom,
    ! [A9: $tType] : semilattice_sup(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_197,axiom,
    ! [A9: $tType] : semilattice_inf(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_198,axiom,
    ! [A9: $tType] : distrib_lattice(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_199,axiom,
    ! [A9: $tType] : bounded_lattice(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_200,axiom,
    ! [A9: $tType] : order_top(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_201,axiom,
    ! [A9: $tType] : order_bot(set(A9)) ).

tff(tcon_Set_Oset___Countable_Ocountable_202,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => countable(set(A9)) ) ).

tff(tcon_Set_Oset___Orderings_Opreorder_203,axiom,
    ! [A9: $tType] : preorder(set(A9)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_204,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => finite_finite(set(A9)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_205,axiom,
    ! [A9: $tType] : lattice(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder_206,axiom,
    ! [A9: $tType] : order(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Otop_207,axiom,
    ! [A9: $tType] : top(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oord_208,axiom,
    ! [A9: $tType] : ord(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Obot_209,axiom,
    ! [A9: $tType] : bot(set(A9)) ).

tff(tcon_Set_Oset___Groups_Ouminus_210,axiom,
    ! [A9: $tType] : uminus(set(A9)) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_211,axiom,
    counta4013691401010221786attice(bool) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_212,axiom,
    condit1219197933456340205attice(bool) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_213,axiom,
    counta3822494911875563373attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_214,axiom,
    comple592849572758109894attice(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_215,axiom,
    topolo4958980785337419405_space(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_216,axiom,
    topolo1944317154257567458pology(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_217,axiom,
    topolo8865339358273720382pology(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_218,axiom,
    bounde4967611905675639751up_bot(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_219,axiom,
    bounde4346867609351753570nf_top(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_220,axiom,
    comple6319245703460814977attice(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_221,axiom,
    topolo2564578578187576103pology(bool) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_222,axiom,
    boolea8198339166811842893lgebra(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_223,axiom,
    bounded_lattice_bot(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_224,axiom,
    topological_t2_space(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_225,axiom,
    topological_t1_space(bool) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_226,axiom,
    comple9053668089753744459l_ccpo(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_227,axiom,
    semilattice_sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_228,axiom,
    semilattice_inf(bool) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_229,axiom,
    distrib_lattice(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_230,axiom,
    bounded_lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_231,axiom,
    order_top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_232,axiom,
    order_bot(bool) ).

tff(tcon_HOL_Obool___Countable_Ocountable_233,axiom,
    countable(bool) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_234,axiom,
    preorder(bool) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_235,axiom,
    linorder(bool) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_236,axiom,
    finite_finite(bool) ).

tff(tcon_HOL_Obool___Lattices_Olattice_237,axiom,
    lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder_238,axiom,
    order(bool) ).

tff(tcon_HOL_Obool___Orderings_Otop_239,axiom,
    top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oord_240,axiom,
    ord(bool) ).

tff(tcon_HOL_Obool___Orderings_Obot_241,axiom,
    bot(bool) ).

tff(tcon_HOL_Obool___Groups_Ouminus_242,axiom,
    uminus(bool) ).

tff(tcon_List_Olist___Countable_Ocountable_243,axiom,
    ! [A9: $tType] :
      ( countable(A9)
     => countable(list(A9)) ) ).

tff(tcon_List_Olist___Nat_Osize_244,axiom,
    ! [A9: $tType] : size(list(A9)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_245,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_246,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_247,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_248,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_249,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_250,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_251,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_252,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_253,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_254,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_255,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_256,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_257,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_258,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_259,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_260,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_261,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_262,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_263,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_264,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_265,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_266,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_267,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_268,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_269,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_270,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_271,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_272,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_273,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_274,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_275,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_276,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_277,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_278,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_279,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_280,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_281,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_282,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_283,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_284,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_285,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_286,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_287,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_288,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_289,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_290,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_291,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_292,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_293,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_294,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_295,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_296,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_297,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_298,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_299,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_300,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_301,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_302,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_303,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_304,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_305,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_306,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_307,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_308,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_309,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_310,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_311,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_312,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_313,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_314,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_315,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_316,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_317,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_318,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_319,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_320,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_321,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_322,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_323,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_324,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_325,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_326,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_327,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_328,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_329,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_330,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_331,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_332,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_333,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_334,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_335,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_336,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_337,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_338,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_339,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_340,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_341,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_342,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_343,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_344,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_345,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_346,axiom,
    dvd(real) ).

tff(tcon_Sum__Type_Osum___Countable_Ocountable_347,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( countable(A9)
        & countable(A20) )
     => countable(sum_sum(A9,A20)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_348,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( finite_finite(A9)
        & finite_finite(A20) )
     => finite_finite(sum_sum(A9,A20)) ) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_349,axiom,
    ! [A9: $tType,A20: $tType] : size(sum_sum(A9,A20)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_350,axiom,
    ! [A9: $tType] : condit1219197933456340205attice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_351,axiom,
    ! [A9: $tType] : counta3822494911875563373attice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_352,axiom,
    ! [A9: $tType] : bounde4967611905675639751up_bot(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_353,axiom,
    ! [A9: $tType] : bounde4346867609351753570nf_top(filter(A9)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_354,axiom,
    ! [A9: $tType] : comple6319245703460814977attice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_355,axiom,
    ! [A9: $tType] : bounded_lattice_bot(filter(A9)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_356,axiom,
    ! [A9: $tType] : comple9053668089753744459l_ccpo(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_357,axiom,
    ! [A9: $tType] : semilattice_sup(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_358,axiom,
    ! [A9: $tType] : semilattice_inf(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_359,axiom,
    ! [A9: $tType] : distrib_lattice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_360,axiom,
    ! [A9: $tType] : bounded_lattice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_361,axiom,
    ! [A9: $tType] : order_top(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_362,axiom,
    ! [A9: $tType] : order_bot(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_363,axiom,
    ! [A9: $tType] : preorder(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_364,axiom,
    ! [A9: $tType] : lattice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_365,axiom,
    ! [A9: $tType] : order(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_366,axiom,
    ! [A9: $tType] : top(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_367,axiom,
    ! [A9: $tType] : ord(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_368,axiom,
    ! [A9: $tType] : bot(filter(A9)) ).

tff(tcon_Option_Ooption___Countable_Ocountable_369,axiom,
    ! [A9: $tType] :
      ( countable(A9)
     => countable(option(A9)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_370,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => finite_finite(option(A9)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_371,axiom,
    ! [A9: $tType] : size(option(A9)) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_372,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_373,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_374,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_375,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_376,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_377,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_378,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_379,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_380,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_381,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_382,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_383,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_384,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_385,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_386,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_387,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_388,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_389,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_390,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_391,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_392,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_393,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_394,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_395,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_396,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_397,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_398,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_399,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_400,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_401,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_402,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_403,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_404,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_405,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_406,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_407,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_408,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_409,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_410,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_411,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_412,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_413,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_414,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_415,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_416,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_417,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_418,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_419,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_420,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_421,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_422,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_423,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_424,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_425,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_426,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_427,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_428,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_429,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_430,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_431,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_432,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_433,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_434,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_435,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_436,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_437,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_438,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_439,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_440,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_441,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_442,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_443,axiom,
    counta4013691401010221786attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_444,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_445,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_446,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_447,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_448,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_449,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_450,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_451,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_452,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_453,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_454,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_455,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_456,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_457,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_458,axiom,
    bounded_lattice_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_459,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_460,axiom,
    comple9053668089753744459l_ccpo(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_461,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_462,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_463,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_464,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_465,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_466,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_467,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_468,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_469,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_470,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_471,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_472,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_473,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_474,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_475,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_476,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_477,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_478,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_479,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable_Ocountable_480,axiom,
    countable(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_481,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_482,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_483,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_484,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_485,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_486,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_487,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_488,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_489,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_490,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_491,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_492,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_493,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_494,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_495,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_496,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_497,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_498,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_499,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_500,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( topolo4958980785337419405_space(A9)
        & topolo4958980785337419405_space(A20) )
     => topolo4958980785337419405_space(product_prod(A9,A20)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_501,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( topological_t2_space(A9)
        & topological_t2_space(A20) )
     => topological_t2_space(product_prod(A9,A20)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_502,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( topological_t1_space(A9)
        & topological_t1_space(A20) )
     => topological_t1_space(product_prod(A9,A20)) ) ).

tff(tcon_Product__Type_Oprod___Countable_Ocountable_503,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( countable(A9)
        & countable(A20) )
     => countable(product_prod(A9,A20)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_504,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( finite_finite(A9)
        & finite_finite(A20) )
     => finite_finite(product_prod(A9,A20)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_505,axiom,
    ! [A9: $tType,A20: $tType] : size(product_prod(A9,A20)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_506,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_507,axiom,
    counta4013691401010221786attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_508,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_509,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_510,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_511,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_512,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_513,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_514,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_515,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_516,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_517,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_518,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_519,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_520,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_521,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_522,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_523,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_524,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable_Ocountable_525,axiom,
    countable(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_526,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_527,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_528,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_529,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_530,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_531,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_532,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_533,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_534,axiom,
    uminus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_535,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_536,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_537,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_538,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_539,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_540,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_541,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_542,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_543,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_544,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_545,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_546,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_547,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_548,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_549,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_550,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_551,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_552,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_553,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_554,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_555,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_556,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_557,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_558,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_559,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_560,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_561,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_562,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_563,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_564,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_565,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_566,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_567,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_568,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_569,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_570,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_571,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_572,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_573,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_574,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_575,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_576,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_577,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_578,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_579,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_580,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_581,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_582,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_583,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_584,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_585,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_586,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_587,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_588,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_589,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_590,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_591,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_592,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_593,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_594,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_595,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_596,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_597,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_598,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_599,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_600,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_601,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_602,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_603,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_604,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_605,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_606,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_607,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_608,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_609,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_610,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_611,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_612,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_613,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_614,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_615,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_616,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_617,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_618,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_619,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_620,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_621,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_622,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_623,axiom,
    dvd(code_integer) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_624,axiom,
    size(vEBT_VEBT) ).

% Helper facts (24)
tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fFalse,X,Y) = Y ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fTrue,X,Y) = X ).

tff(help_fEx_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(aa(A,bool,P,X))
      | pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(fAll(A,P))
      | pp(aa(A,bool,P,X)) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa(bool,bool,fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa(bool,bool,fNot,P))
      | ~ pp(P) ) ).

tff(help_COMBB_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(B,C),Q: fun(A,B),R: A] : aa(A,C,combb(B,C,A,P,Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).

tff(help_COMBC_1_1_U,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,fun(B,C)),Q: B,R: A] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).

tff(help_COMBS_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,fun(B,C)),Q: fun(A,B),R: A] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(fconj(P,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fdisj(P,Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(fdisj(P,Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(fdisj(P,Q)) ) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(A,bool,P,fChoice(A,P)) = aa(fun(A,bool),bool,fEx(A),P) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    vEBT_invar_vebt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,na)),na) ).

%------------------------------------------------------------------------------